| L(s) = 1 | − 8·7-s − 4·17-s + 12·23-s + 8·25-s − 8·31-s + 4·41-s + 36·49-s − 28·71-s − 8·73-s + 40·79-s − 20·89-s + 40·97-s − 8·103-s + 8·113-s + 32·119-s + 24·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 96·161-s + 163-s + 167-s + 48·169-s + ⋯ |
| L(s) = 1 | − 3.02·7-s − 0.970·17-s + 2.50·23-s + 8/5·25-s − 1.43·31-s + 0.624·41-s + 36/7·49-s − 3.32·71-s − 0.936·73-s + 4.50·79-s − 2.11·89-s + 4.06·97-s − 0.788·103-s + 0.752·113-s + 2.93·119-s + 2.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 7.56·161-s + 0.0783·163-s + 0.0773·167-s + 3.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.263886358\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.263886358\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 + T )^{8} \) |
| good | 5 | \( 1 - 8 T^{2} + 16 T^{4} - 168 T^{6} + 1694 T^{8} - 168 p^{2} T^{10} + 16 p^{4} T^{12} - 8 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 24 T^{2} + 592 T^{4} - 8312 T^{6} + 114206 T^{8} - 8312 p^{2} T^{10} + 592 p^{4} T^{12} - 24 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 48 T^{2} + 1340 T^{4} - 26704 T^{6} + 396198 T^{8} - 26704 p^{2} T^{10} + 1340 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 2 T + 38 T^{2} + 70 T^{3} + 706 T^{4} + 70 p T^{5} + 38 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 64 T^{2} + 2012 T^{4} - 42432 T^{6} + 793446 T^{8} - 42432 p^{2} T^{10} + 2012 p^{4} T^{12} - 64 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( ( 1 - 6 T + 74 T^{2} - 334 T^{3} + 2282 T^{4} - 334 p T^{5} + 74 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 29 | \( 1 - 16 T^{2} + 2876 T^{4} - 34928 T^{6} + 3439654 T^{8} - 34928 p^{2} T^{10} + 2876 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) |
| 31 | \( ( 1 + 4 T + 80 T^{2} + 244 T^{3} + 3294 T^{4} + 244 p T^{5} + 80 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( 1 - 192 T^{2} + 18716 T^{4} - 1175872 T^{6} + 51538086 T^{8} - 1175872 p^{2} T^{10} + 18716 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 2 T + 134 T^{2} - 214 T^{3} + 7618 T^{4} - 214 p T^{5} + 134 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 168 T^{2} + 15676 T^{4} - 1026520 T^{6} + 50753126 T^{8} - 1026520 p^{2} T^{10} + 15676 p^{4} T^{12} - 168 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 44 T^{2} - 128 T^{3} + 3302 T^{4} - 128 p T^{5} + 44 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 336 T^{2} + 52604 T^{4} - 5027376 T^{6} + 321653350 T^{8} - 5027376 p^{2} T^{10} + 52604 p^{4} T^{12} - 336 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( ( 1 - 102 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( 1 - 272 T^{2} + 35516 T^{4} - 3030896 T^{6} + 201654822 T^{8} - 3030896 p^{2} T^{10} + 35516 p^{4} T^{12} - 272 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( 1 - 344 T^{2} + 59996 T^{4} - 6783272 T^{6} + 536949606 T^{8} - 6783272 p^{2} T^{10} + 59996 p^{4} T^{12} - 344 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 14 T + 194 T^{2} + 1686 T^{3} + 14330 T^{4} + 1686 p T^{5} + 194 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + 4 T + 92 T^{2} - 580 T^{3} + 166 T^{4} - 580 p T^{5} + 92 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 20 T + 340 T^{2} - 3972 T^{3} + 38678 T^{4} - 3972 p T^{5} + 340 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( 1 - 248 T^{2} + 532 p T^{4} - 5204488 T^{6} + 499369126 T^{8} - 5204488 p^{2} T^{10} + 532 p^{5} T^{12} - 248 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 10 T + 198 T^{2} + 494 T^{3} + 13122 T^{4} + 494 p T^{5} + 198 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 20 T + 300 T^{2} - 2572 T^{3} + 25574 T^{4} - 2572 p T^{5} + 300 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.77156675128064471165304416738, −3.68447421205840334847315931074, −3.66367879380895550598948489794, −3.33331824525136858158142700221, −3.32883985013107710071822543379, −3.17737185504720546932259692444, −3.12283678709362070655620515940, −2.99068755226240933115283423841, −2.93688202775279955240539772746, −2.79855290130210292508832668286, −2.64949067094217903164748436418, −2.38436653837675331480194855860, −2.36695148698659267718687781819, −2.24344426066357262656921425869, −2.11365124213734510672181314321, −1.98600970202194118210003916209, −1.68808477764924519034846881587, −1.39804446712441814067909991757, −1.39266932862749874051297797293, −1.28173118716014851472324709207, −0.873376450381500187089627446108, −0.854943387791684233019425816412, −0.53504506513507596643477098821, −0.41692222181428674877801658412, −0.19930288789041149113167585121,
0.19930288789041149113167585121, 0.41692222181428674877801658412, 0.53504506513507596643477098821, 0.854943387791684233019425816412, 0.873376450381500187089627446108, 1.28173118716014851472324709207, 1.39266932862749874051297797293, 1.39804446712441814067909991757, 1.68808477764924519034846881587, 1.98600970202194118210003916209, 2.11365124213734510672181314321, 2.24344426066357262656921425869, 2.36695148698659267718687781819, 2.38436653837675331480194855860, 2.64949067094217903164748436418, 2.79855290130210292508832668286, 2.93688202775279955240539772746, 2.99068755226240933115283423841, 3.12283678709362070655620515940, 3.17737185504720546932259692444, 3.32883985013107710071822543379, 3.33331824525136858158142700221, 3.66367879380895550598948489794, 3.68447421205840334847315931074, 3.77156675128064471165304416738
Plot not available for L-functions of degree greater than 10.
