| L(s) = 1 | − 5-s + 3.41·7-s + 11-s − 1.41·13-s − 4.24·17-s + 6.82·19-s + 2.82·23-s + 25-s − 3.65·29-s − 0.828·31-s − 3.41·35-s + 7.65·37-s + 7.65·41-s − 5.07·43-s − 12.4·47-s + 4.65·49-s + 10.4·53-s − 55-s − 3.17·59-s + 3.17·61-s + 1.41·65-s + 8.48·67-s + 6.48·71-s − 7.07·73-s + 3.41·77-s + 16.4·79-s − 1.75·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.29·7-s + 0.301·11-s − 0.392·13-s − 1.02·17-s + 1.56·19-s + 0.589·23-s + 0.200·25-s − 0.679·29-s − 0.148·31-s − 0.577·35-s + 1.25·37-s + 1.19·41-s − 0.773·43-s − 1.82·47-s + 0.665·49-s + 1.44·53-s − 0.134·55-s − 0.412·59-s + 0.406·61-s + 0.175·65-s + 1.03·67-s + 0.769·71-s − 0.827·73-s + 0.389·77-s + 1.85·79-s − 0.192·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.258505388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.258505388\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 3.41T + 7T^{2} \) |
| 13 | \( 1 + 1.41T + 13T^{2} \) |
| 17 | \( 1 + 4.24T + 17T^{2} \) |
| 19 | \( 1 - 6.82T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 + 0.828T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 + 5.07T + 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 3.17T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 - 6.48T + 71T^{2} \) |
| 73 | \( 1 + 7.07T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 1.75T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 3.17T + 97T^{2} \) |
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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
−7.915267519737862949108166486253, −7.22706338004649759920767944024, −6.62579518065423300431740768651, −5.57556387930043359208140976744, −4.98992372658770539455532675687, −4.39770839082835835666110504710, −3.59228961202000058700130130778, −2.63328086337728497804891513776, −1.72935820272930836841945637350, −0.77476658773066623187586469196,
0.77476658773066623187586469196, 1.72935820272930836841945637350, 2.63328086337728497804891513776, 3.59228961202000058700130130778, 4.39770839082835835666110504710, 4.98992372658770539455532675687, 5.57556387930043359208140976744, 6.62579518065423300431740768651, 7.22706338004649759920767944024, 7.915267519737862949108166486253
