| L(s) = 1 | − 3·7-s − 9-s + 2·11-s + 9·13-s + 6·17-s − 13·19-s − 2·23-s + 11·29-s − 4·31-s − 12·37-s − 2·41-s − 11·43-s − 7·47-s − 6·49-s − 2·53-s − 7·59-s − 4·61-s + 3·63-s − 14·67-s + 15·71-s + 22·73-s − 6·77-s − 5·79-s − 8·81-s + 18·83-s + 8·89-s − 27·91-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 1/3·9-s + 0.603·11-s + 2.49·13-s + 1.45·17-s − 2.98·19-s − 0.417·23-s + 2.04·29-s − 0.718·31-s − 1.97·37-s − 0.312·41-s − 1.67·43-s − 1.02·47-s − 6/7·49-s − 0.274·53-s − 0.911·59-s − 0.512·61-s + 0.377·63-s − 1.71·67-s + 1.78·71-s + 2.57·73-s − 0.683·77-s − 0.562·79-s − 8/9·81-s + 1.97·83-s + 0.847·89-s − 2.83·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 9 T + 45 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_4$ | \( 1 + 13 T + 79 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 3 p T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 105 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 2 T + 38 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 11 T + 115 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 7 T + 95 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 2 T + 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 7 T + 129 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 15 T + 197 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 103 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 3 T - 85 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.52067240281180153504138996552, −6.91132539108663668359038972173, −6.51262923171420901196638422480, −6.41221038653902978000855724354, −6.25070110364344374181937364120, −6.20391333795979263369329483070, −5.30566257371263762894874685016, −5.22113857809169481961566477702, −4.77868494317365790211205999089, −4.19112495100410277533944944911, −3.77260860953208103747135413723, −3.68850670177880344444030869434, −3.29401093042483940489595713074, −3.00846747387846470259935183423, −2.38288952277602740832530962922, −1.83317060212384641520450692974, −1.41929803902827497715119417380, −1.11637705851755774626304151633, 0, 0,
1.11637705851755774626304151633, 1.41929803902827497715119417380, 1.83317060212384641520450692974, 2.38288952277602740832530962922, 3.00846747387846470259935183423, 3.29401093042483940489595713074, 3.68850670177880344444030869434, 3.77260860953208103747135413723, 4.19112495100410277533944944911, 4.77868494317365790211205999089, 5.22113857809169481961566477702, 5.30566257371263762894874685016, 6.20391333795979263369329483070, 6.25070110364344374181937364120, 6.41221038653902978000855724354, 6.51262923171420901196638422480, 6.91132539108663668359038972173, 7.52067240281180153504138996552
