| L(s) = 1 | + 5·3-s − 5·5-s + 6·7-s + 13·9-s + 6·11-s − 7·13-s − 25·15-s − 8·17-s − 9·19-s + 30·21-s + 9·23-s + 10·25-s + 30·27-s − 3·29-s + 13·31-s + 30·33-s − 30·35-s + 37-s − 35·39-s + 16·41-s + 22·43-s − 65·45-s + 4·47-s − 3·49-s − 40·51-s − 19·53-s − 30·55-s + ⋯ |
| L(s) = 1 | + 2.88·3-s − 2.23·5-s + 2.26·7-s + 13/3·9-s + 1.80·11-s − 1.94·13-s − 6.45·15-s − 1.94·17-s − 2.06·19-s + 6.54·21-s + 1.87·23-s + 2·25-s + 5.77·27-s − 0.557·29-s + 2.33·31-s + 5.22·33-s − 5.07·35-s + 0.164·37-s − 5.60·39-s + 2.49·41-s + 3.35·43-s − 9.68·45-s + 0.583·47-s − 3/7·49-s − 5.60·51-s − 2.60·53-s − 4.04·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.640147114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.640147114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | $C_4$ | \( 1 + p T + 3 p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_4\times C_2$ | \( 1 - 5 T + 4 p T^{2} - 25 T^{3} + 49 T^{4} - 25 p T^{5} + 4 p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 3 T + 15 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2:C_4$ | \( 1 - 6 T + 5 T^{2} + 6 T^{3} + 49 T^{4} + 6 p T^{5} + 5 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 7 T + 11 T^{2} - 79 T^{3} - 536 T^{4} - 79 p T^{5} + 11 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 8 T + 7 T^{2} - 110 T^{3} - 579 T^{4} - 110 p T^{5} + 7 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_4\times C_2$ | \( 1 + 9 T + 27 T^{2} - 83 T^{3} - 960 T^{4} - 83 p T^{5} + 27 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2:C_4$ | \( 1 - 9 T + 38 T^{2} - 255 T^{3} + 1741 T^{4} - 255 p T^{5} + 38 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 3 T + 5 T^{2} - 3 p T^{3} - 4 p T^{4} - 3 p^{2} T^{5} + 5 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 13 T + 108 T^{2} - 821 T^{3} + 5525 T^{4} - 821 p T^{5} + 108 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - T + 14 T^{2} + 103 T^{3} + 739 T^{4} + 103 p T^{5} + 14 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 16 T + 95 T^{2} - 534 T^{3} + 4109 T^{4} - 534 p T^{5} + 95 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 4 T - T^{2} - 318 T^{3} + 3479 T^{4} - 318 p T^{5} - p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 19 T + 188 T^{2} + 1765 T^{3} + 15251 T^{4} + 1765 p T^{5} + 188 p^{2} T^{6} + 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 6 T - 43 T^{2} - 402 T^{3} + 475 T^{4} - 402 p T^{5} - 43 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 23 T + 318 T^{2} + 3511 T^{3} + 31355 T^{4} + 3511 p T^{5} + 318 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 22 T + 117 T^{2} - 970 T^{3} - 16399 T^{4} - 970 p T^{5} + 117 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 10 T - 11 T^{2} + 790 T^{3} - 6489 T^{4} + 790 p T^{5} - 11 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_4\times C_2$ | \( 1 + 11 T + 48 T^{2} - 275 T^{3} - 6529 T^{4} - 275 p T^{5} + 48 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 15 T + 111 T^{2} - 1495 T^{3} + 19536 T^{4} - 1495 p T^{5} + 111 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + T + 58 T^{2} + 335 T^{3} + 7041 T^{4} + 335 p T^{5} + 58 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_4\times C_2$ | \( 1 + 4 T - 73 T^{2} - 648 T^{3} + 3905 T^{4} - 648 p T^{5} - 73 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 36 T + 749 T^{2} - 11172 T^{3} + 124789 T^{4} - 11172 p T^{5} + 749 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.168205984833807359921736757614, −8.859993352833541823017662914576, −8.613459789247073535035353261223, −8.230891427521361793792822089441, −7.969011993015351599427181722138, −7.963095905909545202809939208296, −7.62136861894541484636262768885, −7.45746876674645464331543824589, −7.38688724523140684504483379588, −6.77508143638941654374302885484, −6.59829440124508259258889095091, −6.16775937932023986386626557286, −6.07248932314260346534353139734, −4.81081477263425944096717602594, −4.79085624826043896341536773143, −4.57583726517368218232392400119, −4.46887371405670977437919867622, −4.13023400979335061773916604682, −4.04559620663862811180051570608, −3.17370587119169730974811346610, −2.97578655875082507004543226993, −2.79818339854156340071020354116, −2.03789560009096921956091206354, −2.02488307079206194451136367777, −1.12208157242451914798598511815,
1.12208157242451914798598511815, 2.02488307079206194451136367777, 2.03789560009096921956091206354, 2.79818339854156340071020354116, 2.97578655875082507004543226993, 3.17370587119169730974811346610, 4.04559620663862811180051570608, 4.13023400979335061773916604682, 4.46887371405670977437919867622, 4.57583726517368218232392400119, 4.79085624826043896341536773143, 4.81081477263425944096717602594, 6.07248932314260346534353139734, 6.16775937932023986386626557286, 6.59829440124508259258889095091, 6.77508143638941654374302885484, 7.38688724523140684504483379588, 7.45746876674645464331543824589, 7.62136861894541484636262768885, 7.963095905909545202809939208296, 7.969011993015351599427181722138, 8.230891427521361793792822089441, 8.613459789247073535035353261223, 8.859993352833541823017662914576, 9.168205984833807359921736757614
