| L(s) = 1 | − 6·2-s + 18·4-s − 6·5-s + 26·7-s − 36·8-s + 36·10-s − 28·13-s − 156·14-s + 52·16-s − 108·20-s − 60·23-s + 34·25-s + 168·26-s + 468·28-s − 120·29-s + 148·31-s − 156·35-s + 20·37-s + 216·40-s + 2·43-s + 360·46-s − 6·47-s + 338·49-s − 204·50-s − 504·52-s − 936·56-s + 720·58-s + ⋯ |
| L(s) = 1 | − 3·2-s + 9/2·4-s − 6/5·5-s + 26/7·7-s − 9/2·8-s + 18/5·10-s − 2.15·13-s − 11.1·14-s + 13/4·16-s − 5.39·20-s − 2.60·23-s + 1.35·25-s + 6.46·26-s + 16.7·28-s − 4.13·29-s + 4.77·31-s − 4.45·35-s + 0.540·37-s + 27/5·40-s + 2/43·43-s + 7.82·46-s − 0.127·47-s + 6.89·49-s − 4.07·50-s − 9.69·52-s − 16.7·56-s + 12.4·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{32} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.01613511745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01613511745\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 6 T + 2 T^{2} - 12 p T^{3} - 9 p^{2} T^{4} - 12 p^{3} T^{5} + 2 p^{4} T^{6} + 6 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 2 | \( ( 1 + p T + p T^{2} + p^{3} T^{3} + p^{4} T^{4} )^{2}( 1 + p T + p T^{2} - 3 p^{2} T^{3} - 7 p^{2} T^{4} - 3 p^{4} T^{5} + p^{5} T^{6} + p^{7} T^{7} + p^{8} T^{8} ) \) |
| 7 | \( 1 - 26 T + 338 T^{2} - 2392 T^{3} + 902 p T^{4} + 51038 T^{5} - 600288 T^{6} + 2518854 T^{7} - 7988669 T^{8} + 2518854 p^{2} T^{9} - 600288 p^{4} T^{10} + 51038 p^{6} T^{11} + 902 p^{9} T^{12} - 2392 p^{10} T^{13} + 338 p^{12} T^{14} - 26 p^{14} T^{15} + p^{16} T^{16} \) |
| 11 | \( 1 - 34 p T^{2} + 78649 T^{4} - 1086130 p T^{6} + 1486462132 T^{8} - 1086130 p^{5} T^{10} + 78649 p^{8} T^{12} - 34 p^{13} T^{14} + p^{16} T^{16} \) |
| 13 | \( 1 + 28 T + 392 T^{2} - 1624 T^{3} - 120850 T^{4} - 1960084 T^{5} - 6190464 T^{6} + 232029588 T^{7} + 5282655283 T^{8} + 232029588 p^{2} T^{9} - 6190464 p^{4} T^{10} - 1960084 p^{6} T^{11} - 120850 p^{8} T^{12} - 1624 p^{10} T^{13} + 392 p^{12} T^{14} + 28 p^{14} T^{15} + p^{16} T^{16} \) |
| 17 | \( 1 + 23288 T^{4} - 108642 p^{4} T^{8} + 23288 p^{8} T^{12} + p^{16} T^{16} \) |
| 19 | \( ( 1 - 778 T^{2} + 303099 T^{4} - 778 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 + 60 T + 1800 T^{2} + 36000 T^{3} + 340178 T^{4} + 2653980 T^{5} + 194918400 T^{6} + 9729805020 T^{7} + 300980936403 T^{8} + 9729805020 p^{2} T^{9} + 194918400 p^{4} T^{10} + 2653980 p^{6} T^{11} + 340178 p^{8} T^{12} + 36000 p^{10} T^{13} + 1800 p^{12} T^{14} + 60 p^{14} T^{15} + p^{16} T^{16} \) |
| 29 | \( ( 1 + 60 T + 2615 T^{2} + 84900 T^{3} + 2304144 T^{4} + 84900 p^{2} T^{5} + 2615 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 31 | \( ( 1 - 37 T + 408 T^{2} - 37 p^{2} T^{3} + p^{4} T^{4} )^{4} \) |
| 37 | \( ( 1 - 10 T + 50 T^{2} + 32490 T^{3} - 213922 T^{4} + 32490 p^{2} T^{5} + 50 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 5174 T^{2} + 15018889 T^{4} - 31560701510 T^{6} + 54463769314612 T^{8} - 31560701510 p^{4} T^{10} + 15018889 p^{8} T^{12} - 5174 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( 1 - 2 T + 2 T^{2} + 2312 T^{3} + 940106 T^{4} - 5913130 T^{5} + 12618720 T^{6} - 11556516690 T^{7} - 10751970804125 T^{8} - 11556516690 p^{2} T^{9} + 12618720 p^{4} T^{10} - 5913130 p^{6} T^{11} + 940106 p^{8} T^{12} + 2312 p^{10} T^{13} + 2 p^{12} T^{14} - 2 p^{14} T^{15} + p^{16} T^{16} \) |
| 47 | \( 1 + 6 T + 18 T^{2} + 36 T^{3} - 4874758 T^{4} - 21399942 T^{5} - 40653360 T^{6} - 17308487982 T^{7} - 47272777101 T^{8} - 17308487982 p^{2} T^{9} - 40653360 p^{4} T^{10} - 21399942 p^{6} T^{11} - 4874758 p^{8} T^{12} + 36 p^{10} T^{13} + 18 p^{12} T^{14} + 6 p^{14} T^{15} + p^{16} T^{16} \) |
| 53 | \( 1 - 23370376 T^{4} + 249103570345566 T^{8} - 23370376 p^{8} T^{12} + p^{16} T^{16} \) |
| 59 | \( ( 1 - 264 T + 35435 T^{2} - 3221592 T^{3} + 217666440 T^{4} - 3221592 p^{2} T^{5} + 35435 p^{4} T^{6} - 264 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 61 | \( ( 1 - 56 T - 4334 T^{2} - 1568 T^{3} + 30286003 T^{4} - 1568 p^{2} T^{5} - 4334 p^{4} T^{6} - 56 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( 1 + 76 T + 2888 T^{2} - 470440 T^{3} - 59079262 T^{4} - 3717986956 T^{5} - 1289203200 T^{6} + 16566773334084 T^{7} + 1634293985113843 T^{8} + 16566773334084 p^{2} T^{9} - 1289203200 p^{4} T^{10} - 3717986956 p^{6} T^{11} - 59079262 p^{8} T^{12} - 470440 p^{10} T^{13} + 2888 p^{12} T^{14} + 76 p^{14} T^{15} + p^{16} T^{16} \) |
| 71 | \( ( 1 + 14894 T^{2} + 105189507 T^{4} + 14894 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 73 | \( ( 1 - 130 T + 8450 T^{2} - 955110 T^{3} + 103884494 T^{4} - 955110 p^{2} T^{5} + 8450 p^{4} T^{6} - 130 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( 1 + 8344 T^{2} - 3577214 T^{4} - 39221906528 T^{6} + 1129269683686723 T^{8} - 39221906528 p^{4} T^{10} - 3577214 p^{8} T^{12} + 8344 p^{12} T^{14} + p^{16} T^{16} \) |
| 83 | \( 1 + 582 T + 169362 T^{2} + 32856228 T^{3} + 4870379162 T^{4} + 603331449450 T^{5} + 66047607139248 T^{6} + 6505169065814274 T^{7} + 572797621325593395 T^{8} + 6505169065814274 p^{2} T^{9} + 66047607139248 p^{4} T^{10} + 603331449450 p^{6} T^{11} + 4870379162 p^{8} T^{12} + 32856228 p^{10} T^{13} + 169362 p^{12} T^{14} + 582 p^{14} T^{15} + p^{16} T^{16} \) |
| 89 | \( ( 1 - 22054 T^{2} + 242479707 T^{4} - 22054 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 97 | \( 1 + 274 T + 37538 T^{2} - 7672 T^{3} - 618781366 T^{4} - 96901498222 T^{5} - 3323166166128 T^{6} + 682503166725474 T^{7} + 124863776311350211 T^{8} + 682503166725474 p^{2} T^{9} - 3323166166128 p^{4} T^{10} - 96901498222 p^{6} T^{11} - 618781366 p^{8} T^{12} - 7672 p^{10} T^{13} + 37538 p^{12} T^{14} + 274 p^{14} T^{15} + p^{16} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.73061638368663697427928592445, −4.39658666094321895612929719344, −4.33983232650743817820549093129, −4.29545854681320105534474422525, −4.19740496064627822230719113664, −4.01277919356020687201328816274, −3.99328708832233186630052108184, −3.85486342409344142985793781887, −3.70527956907606059025802738500, −3.11702705520487763645299696072, −3.05405141495602659540426795914, −2.88020029934054326146090200768, −2.69028271068019239420185079417, −2.42267836831733997022443065919, −2.25102994575018287014276332514, −2.24082499834604780497235344422, −2.03288560575056543032877602810, −2.01935910358199751547208613987, −1.58365873013816416772492794754, −1.28685079925716370266107063566, −1.18898223047597777175536420849, −0.887178396082226016697725193923, −0.816456329695651648145498841472, −0.48433965363516276154448771566, −0.02923640080543809858662274547,
0.02923640080543809858662274547, 0.48433965363516276154448771566, 0.816456329695651648145498841472, 0.887178396082226016697725193923, 1.18898223047597777175536420849, 1.28685079925716370266107063566, 1.58365873013816416772492794754, 2.01935910358199751547208613987, 2.03288560575056543032877602810, 2.24082499834604780497235344422, 2.25102994575018287014276332514, 2.42267836831733997022443065919, 2.69028271068019239420185079417, 2.88020029934054326146090200768, 3.05405141495602659540426795914, 3.11702705520487763645299696072, 3.70527956907606059025802738500, 3.85486342409344142985793781887, 3.99328708832233186630052108184, 4.01277919356020687201328816274, 4.19740496064627822230719113664, 4.29545854681320105534474422525, 4.33983232650743817820549093129, 4.39658666094321895612929719344, 4.73061638368663697427928592445
Plot not available for L-functions of degree greater than 10.
