| L(s) = 1 | − 3·2-s − 6·3-s + 2·4-s + 18·6-s + 4·7-s + 3·8-s + 15·9-s + 7·11-s − 12·12-s + 5·13-s − 12·14-s − 6·16-s − 5·17-s − 45·18-s + 8·19-s − 24·21-s − 21·22-s + 4·23-s − 18·24-s − 15·26-s − 17·27-s + 8·28-s − 2·29-s + 6·32-s − 42·33-s + 15·34-s + 30·36-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 3.46·3-s + 4-s + 7.34·6-s + 1.51·7-s + 1.06·8-s + 5·9-s + 2.11·11-s − 3.46·12-s + 1.38·13-s − 3.20·14-s − 3/2·16-s − 1.21·17-s − 10.6·18-s + 1.83·19-s − 5.23·21-s − 4.47·22-s + 0.834·23-s − 3.67·24-s − 2.94·26-s − 3.27·27-s + 1.51·28-s − 0.371·29-s + 1.06·32-s − 7.31·33-s + 2.57·34-s + 5·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 23 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 3 T + 7 T^{2} + 3 p^{2} T^{3} + 19 T^{4} + 3 p^{3} T^{5} + 7 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 2 p T + 7 p T^{2} + 53 T^{3} + 104 T^{4} + 53 p T^{5} + 7 p^{3} T^{6} + 2 p^{4} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 7 T + 58 T^{2} - 236 T^{3} + 1030 T^{4} - 236 p T^{5} + 58 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 5 T + 4 p T^{2} - 190 T^{3} + 1012 T^{4} - 190 p T^{5} + 4 p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 5 T + 56 T^{2} + 216 T^{3} + 1304 T^{4} + 216 p T^{5} + 56 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 8 T + 73 T^{2} - 331 T^{3} + 1840 T^{4} - 331 p T^{5} + 73 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 2 T + 80 T^{2} + 222 T^{3} + 2942 T^{4} + 222 p T^{5} + 80 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 75 T^{2} + 117 T^{3} + 2696 T^{4} + 117 p T^{5} + 75 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 13 T + 178 T^{2} + 1292 T^{3} + 9928 T^{4} + 1292 p T^{5} + 178 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - T + 121 T^{2} - 242 T^{3} + 6474 T^{4} - 242 p T^{5} + 121 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 14 T + 161 T^{2} + 1421 T^{3} + 9720 T^{4} + 1421 p T^{5} + 161 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 4 T + 109 T^{2} + 53 T^{3} + 5092 T^{4} + 53 p T^{5} + 109 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - T + 202 T^{2} - 146 T^{3} + 15792 T^{4} - 146 p T^{5} + 202 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 7 T + 162 T^{2} + 752 T^{3} + 12318 T^{4} + 752 p T^{5} + 162 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 7 T + 190 T^{2} + 984 T^{3} + 15464 T^{4} + 984 p T^{5} + 190 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 15 T + 182 T^{2} + 1634 T^{3} + 14382 T^{4} + 1634 p T^{5} + 182 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 3 p T^{2} - 199 T^{3} + 20156 T^{4} - 199 p T^{5} + 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 3 T + 122 T^{2} + 960 T^{3} + 8920 T^{4} + 960 p T^{5} + 122 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 14 T + 319 T^{2} + 2683 T^{3} + 35728 T^{4} + 2683 p T^{5} + 319 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 3 T + 180 T^{2} + 1556 T^{3} + 14858 T^{4} + 1556 p T^{5} + 180 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 11 T + 232 T^{2} + 1102 T^{3} + 20340 T^{4} + 1102 p T^{5} + 232 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 9 T + 241 T^{2} + 2090 T^{3} + 31298 T^{4} + 2090 p T^{5} + 241 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.41636428246101766145981204936, −6.39939931787606755698226973461, −5.78499763275147172482646480858, −5.74871819521952856506474359679, −5.57982871134497509854149353775, −5.41054085051735988843654680815, −5.31327085850430989727043274995, −5.07189164491413086782014006466, −4.99622415128668725447751235207, −4.65393103670962178688222764386, −4.36089172186573992515905461349, −4.29603762375378564986972354542, −4.25245487365455191128967705450, −3.69040139942489194656493381627, −3.65600550297488712599264901157, −3.45987766854294797220654808102, −3.04988772516664253131280767091, −2.71452539995752640046035443696, −2.54142120592904068732462123577, −2.03457575611669316733552170143, −1.54881521268127110346560788557, −1.32033693993254496084772057333, −1.22517484436859893682346863186, −1.21876168957629354836119729012, −1.13811787906921475872862509639, 0, 0, 0, 0,
1.13811787906921475872862509639, 1.21876168957629354836119729012, 1.22517484436859893682346863186, 1.32033693993254496084772057333, 1.54881521268127110346560788557, 2.03457575611669316733552170143, 2.54142120592904068732462123577, 2.71452539995752640046035443696, 3.04988772516664253131280767091, 3.45987766854294797220654808102, 3.65600550297488712599264901157, 3.69040139942489194656493381627, 4.25245487365455191128967705450, 4.29603762375378564986972354542, 4.36089172186573992515905461349, 4.65393103670962178688222764386, 4.99622415128668725447751235207, 5.07189164491413086782014006466, 5.31327085850430989727043274995, 5.41054085051735988843654680815, 5.57982871134497509854149353775, 5.74871819521952856506474359679, 5.78499763275147172482646480858, 6.39939931787606755698226973461, 6.41636428246101766145981204936
