| L(s) = 1 | + 2-s − 5-s − 7-s − 8-s − 3·9-s − 10-s − 3·11-s + 7·13-s − 14-s − 16-s − 6·17-s − 3·18-s + 4·19-s − 3·22-s + 6·23-s + 7·26-s − 6·29-s − 2·31-s − 6·34-s + 35-s + 4·37-s + 4·38-s + 40-s − 2·43-s + 3·45-s + 6·46-s + 9·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s − 0.377·7-s − 0.353·8-s − 9-s − 0.316·10-s − 0.904·11-s + 1.94·13-s − 0.267·14-s − 1/4·16-s − 1.45·17-s − 0.707·18-s + 0.917·19-s − 0.639·22-s + 1.25·23-s + 1.37·26-s − 1.11·29-s − 0.359·31-s − 1.02·34-s + 0.169·35-s + 0.657·37-s + 0.648·38-s + 0.158·40-s − 0.304·43-s + 0.447·45-s + 0.884·46-s + 1.31·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 396900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.580728970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.580728970\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 2 T - 27 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−11.02899682959039022957422339945, −10.85159304727180832664473029993, −9.852369508344136494152828533485, −9.278119897006643230124257513161, −9.207459433361387903284111137088, −8.656724238737901460993706924875, −8.044401932808785731129259804198, −7.967294812938959796112940887013, −7.19147285393687959779301365795, −6.69143538748531820733094173730, −6.10311326534115306126471650699, −5.97187231975871790783285480418, −5.16369681359052419291252148112, −4.97560946437625996538807041027, −4.23438658234391449612922307025, −3.57546554545646443228642307645, −3.30255262924032122707808302392, −2.72799219891489375905379303114, −1.86569059496937546483952543249, −0.61233565255700156135411541140,
0.61233565255700156135411541140, 1.86569059496937546483952543249, 2.72799219891489375905379303114, 3.30255262924032122707808302392, 3.57546554545646443228642307645, 4.23438658234391449612922307025, 4.97560946437625996538807041027, 5.16369681359052419291252148112, 5.97187231975871790783285480418, 6.10311326534115306126471650699, 6.69143538748531820733094173730, 7.19147285393687959779301365795, 7.967294812938959796112940887013, 8.044401932808785731129259804198, 8.656724238737901460993706924875, 9.207459433361387903284111137088, 9.278119897006643230124257513161, 9.852369508344136494152828533485, 10.85159304727180832664473029993, 11.02899682959039022957422339945
