| L(s) = 1 | + 2-s − 2·4-s + 4·5-s + 4·7-s − 2·8-s − 3·9-s + 4·10-s − 2·11-s + 4·14-s + 2·16-s − 3·18-s − 18·19-s − 8·20-s − 2·22-s − 12·23-s + 10·25-s − 2·27-s − 8·28-s − 2·29-s − 18·31-s + 3·32-s + 16·35-s + 6·36-s − 4·37-s − 18·38-s − 8·40-s − 12·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 4-s + 1.78·5-s + 1.51·7-s − 0.707·8-s − 9-s + 1.26·10-s − 0.603·11-s + 1.06·14-s + 1/2·16-s − 0.707·18-s − 4.12·19-s − 1.78·20-s − 0.426·22-s − 2.50·23-s + 2·25-s − 0.384·27-s − 1.51·28-s − 0.371·29-s − 3.23·31-s + 0.530·32-s + 2.70·35-s + 36-s − 0.657·37-s − 2.91·38-s − 1.26·40-s − 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 7^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + 3 T^{2} - 3 T^{3} + 5 T^{4} - 3 p T^{5} + 3 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + p T^{2} + 2 T^{3} + 11 T^{4} + 2 p T^{5} + p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 12 T^{2} - 46 T^{3} - 58 T^{4} - 46 p T^{5} + 12 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 27 T^{2} - 50 T^{3} + 503 T^{4} - 50 p T^{5} + 27 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 18 T + 179 T^{2} + 1228 T^{3} + 6201 T^{4} + 1228 p T^{5} + 179 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 12 T + 100 T^{2} + 516 T^{3} + 2662 T^{4} + 516 p T^{5} + 100 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 2 T + 99 T^{2} + 152 T^{3} + 4121 T^{4} + 152 p T^{5} + 99 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 18 T + 227 T^{2} + 1868 T^{3} + 12237 T^{4} + 1868 p T^{5} + 227 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 4 T + 83 T^{2} + 346 T^{3} + 4299 T^{4} + 346 p T^{5} + 83 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 12 T + 137 T^{2} + 1152 T^{3} + 8277 T^{4} + 1152 p T^{5} + 137 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 2 T + 152 T^{2} - 234 T^{3} + 9390 T^{4} - 234 p T^{5} + 152 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 18 T + 260 T^{2} - 2426 T^{3} + 19718 T^{4} - 2426 p T^{5} + 260 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 2 T + 56 T^{2} - 22 T^{3} + 3022 T^{4} - 22 p T^{5} + 56 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 30 T + 555 T^{2} + 6724 T^{3} + 60749 T^{4} + 6724 p T^{5} + 555 p^{2} T^{6} + 30 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 124 T^{2} + 344 T^{3} + 7766 T^{4} + 344 p T^{5} + 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 12 T + 147 T^{2} + 894 T^{3} + 9743 T^{4} + 894 p T^{5} + 147 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 8 T + 144 T^{2} - 680 T^{3} + 11534 T^{4} - 680 p T^{5} + 144 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 34 T + 644 T^{2} + 8342 T^{3} + 81718 T^{4} + 8342 p T^{5} + 644 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 12 T + 361 T^{2} - 2900 T^{3} + 44573 T^{4} - 2900 p T^{5} + 361 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 2 T + 48 T^{2} + 186 T^{3} + 4958 T^{4} + 186 p T^{5} + 48 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 24 T + 397 T^{2} + 4596 T^{3} + 48609 T^{4} + 4596 p T^{5} + 397 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 14 T + 252 T^{2} + 1354 T^{3} + 20070 T^{4} + 1354 p T^{5} + 252 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.91269974681839235543891304745, −5.85625205973902353181546135019, −5.77000578722065633480475240110, −5.50785867846352588800238236155, −5.42811382404055336128955366843, −5.09595600013491178076681993236, −4.96160499480693601013153615683, −4.64740339616410891779384246423, −4.53253213131672204488347636393, −4.48380374224978701737952155046, −4.15523139561266723730287574955, −4.03812954826405486911178847548, −3.97933360002876818089264392198, −3.61092116831687800093752951694, −3.46483915188580897093603918322, −3.13520707085837954800515150810, −2.70624787435248701326030035441, −2.67554616229731431733662132644, −2.39994542709596849934232265432, −2.21238526112423691984981995515, −1.91724200675977308113484494677, −1.82137420971424929788536866400, −1.52149686857869349495238801097, −1.52120488247790183493791305460, −1.11295807328380048286699201921, 0, 0, 0, 0,
1.11295807328380048286699201921, 1.52120488247790183493791305460, 1.52149686857869349495238801097, 1.82137420971424929788536866400, 1.91724200675977308113484494677, 2.21238526112423691984981995515, 2.39994542709596849934232265432, 2.67554616229731431733662132644, 2.70624787435248701326030035441, 3.13520707085837954800515150810, 3.46483915188580897093603918322, 3.61092116831687800093752951694, 3.97933360002876818089264392198, 4.03812954826405486911178847548, 4.15523139561266723730287574955, 4.48380374224978701737952155046, 4.53253213131672204488347636393, 4.64740339616410891779384246423, 4.96160499480693601013153615683, 5.09595600013491178076681993236, 5.42811382404055336128955366843, 5.50785867846352588800238236155, 5.77000578722065633480475240110, 5.85625205973902353181546135019, 5.91269974681839235543891304745
