| L(s) = 1 | + 3-s − 5-s − 4·7-s + 9-s − 2·13-s − 15-s + 2·17-s − 19-s − 4·21-s + 25-s + 27-s + 6·29-s − 8·31-s + 4·35-s − 2·37-s − 2·39-s + 6·41-s − 8·43-s − 45-s + 8·47-s + 9·49-s + 2·51-s + 10·53-s − 57-s + 12·59-s + 2·61-s − 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.229·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.676·35-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s − 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.280·51-s + 1.37·53-s − 0.132·57-s + 1.56·59-s + 0.256·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.87766283303192, −15.68744512267124, −14.87723546833326, −14.54813875318574, −13.83266506829141, −13.25913387426166, −12.79312557989403, −12.26288462490071, −11.86288665701360, −10.94343291036844, −10.34030771647577, −9.852110294904621, −9.355976751480663, −8.722301150775336, −8.206881154234096, −7.307809838265882, −7.064194477012753, −6.350108702318736, −5.622576779192431, −4.909462244334021, −3.912126853355947, −3.631338029744285, −2.801411818131566, −2.268689270581132, −0.9911601774516588, 0,
0.9911601774516588, 2.268689270581132, 2.801411818131566, 3.631338029744285, 3.912126853355947, 4.909462244334021, 5.622576779192431, 6.350108702318736, 7.064194477012753, 7.307809838265882, 8.206881154234096, 8.722301150775336, 9.355976751480663, 9.852110294904621, 10.34030771647577, 10.94343291036844, 11.86288665701360, 12.26288462490071, 12.79312557989403, 13.25913387426166, 13.83266506829141, 14.54813875318574, 14.87723546833326, 15.68744512267124, 15.87766283303192
