| L(s) = 1 | − 5-s + 2.81·7-s + 4.70·11-s + 4.24·13-s − 3.85·17-s + 6.70·19-s − 23-s + 25-s + 1.42·29-s − 9.03·31-s − 2.81·35-s + 5.28·37-s + 10.3·41-s + 12.1·43-s − 10.0·47-s + 0.921·49-s + 9.17·53-s − 4.70·55-s − 6.36·59-s − 10.4·61-s − 4.24·65-s + 6.50·67-s − 8.74·71-s − 0.240·73-s + 13.2·77-s − 4.48·79-s − 13.1·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.06·7-s + 1.41·11-s + 1.17·13-s − 0.934·17-s + 1.53·19-s − 0.208·23-s + 0.200·25-s + 0.264·29-s − 1.62·31-s − 0.475·35-s + 0.868·37-s + 1.62·41-s + 1.85·43-s − 1.46·47-s + 0.131·49-s + 1.25·53-s − 0.634·55-s − 0.828·59-s − 1.33·61-s − 0.525·65-s + 0.794·67-s − 1.03·71-s − 0.0280·73-s + 1.51·77-s − 0.504·79-s − 1.43·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.465386584\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.465386584\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 2.81T + 7T^{2} \) |
| 11 | \( 1 - 4.70T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 3.85T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 29 | \( 1 - 1.42T + 29T^{2} \) |
| 31 | \( 1 + 9.03T + 31T^{2} \) |
| 37 | \( 1 - 5.28T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 10.0T + 47T^{2} \) |
| 53 | \( 1 - 9.17T + 53T^{2} \) |
| 59 | \( 1 + 6.36T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 6.50T + 67T^{2} \) |
| 71 | \( 1 + 8.74T + 71T^{2} \) |
| 73 | \( 1 + 0.240T + 73T^{2} \) |
| 79 | \( 1 + 4.48T + 79T^{2} \) |
| 83 | \( 1 + 13.1T + 83T^{2} \) |
| 89 | \( 1 - 7.62T + 89T^{2} \) |
| 97 | \( 1 + 0.385T + 97T^{2} \) |
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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
−8.429595741301739766188265285990, −7.67473768829531135023731723072, −7.08984765432574458336965612643, −6.16083064387564137213039116286, −5.52690729709112822185883097805, −4.40136731492024895530579149451, −4.03869359143274857887925433193, −3.04652223449902617214032525658, −1.73390245124033855369817568314, −0.990654592544208267476059116202,
0.990654592544208267476059116202, 1.73390245124033855369817568314, 3.04652223449902617214032525658, 4.03869359143274857887925433193, 4.40136731492024895530579149451, 5.52690729709112822185883097805, 6.16083064387564137213039116286, 7.08984765432574458336965612643, 7.67473768829531135023731723072, 8.429595741301739766188265285990
