| L(s) = 1 | − 0.0898·2-s − 3-s − 1.99·4-s + 0.0898·6-s + 4.36·7-s + 0.358·8-s + 9-s + 4.39·11-s + 1.99·12-s − 1.98·13-s − 0.391·14-s + 3.95·16-s + 0.997·17-s − 0.0898·18-s + 1.35·19-s − 4.36·21-s − 0.394·22-s + 2.35·23-s − 0.358·24-s + 0.177·26-s − 27-s − 8.68·28-s − 7.97·29-s − 3.67·31-s − 1.07·32-s − 4.39·33-s − 0.0895·34-s + ⋯ |
| L(s) = 1 | − 0.0635·2-s − 0.577·3-s − 0.995·4-s + 0.0366·6-s + 1.64·7-s + 0.126·8-s + 0.333·9-s + 1.32·11-s + 0.575·12-s − 0.549·13-s − 0.104·14-s + 0.987·16-s + 0.241·17-s − 0.0211·18-s + 0.309·19-s − 0.951·21-s − 0.0840·22-s + 0.491·23-s − 0.0731·24-s + 0.0349·26-s − 0.192·27-s − 1.64·28-s − 1.48·29-s − 0.660·31-s − 0.189·32-s − 0.764·33-s − 0.0153·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.417910174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.417910174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 0.0898T + 2T^{2} \) |
| 7 | \( 1 - 4.36T + 7T^{2} \) |
| 11 | \( 1 - 4.39T + 11T^{2} \) |
| 13 | \( 1 + 1.98T + 13T^{2} \) |
| 17 | \( 1 - 0.997T + 17T^{2} \) |
| 19 | \( 1 - 1.35T + 19T^{2} \) |
| 23 | \( 1 - 2.35T + 23T^{2} \) |
| 29 | \( 1 + 7.97T + 29T^{2} \) |
| 31 | \( 1 + 3.67T + 31T^{2} \) |
| 37 | \( 1 + 1.43T + 37T^{2} \) |
| 41 | \( 1 + 5.98T + 41T^{2} \) |
| 43 | \( 1 - 2.68T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 - 6.68T + 59T^{2} \) |
| 61 | \( 1 + 9.45T + 61T^{2} \) |
| 67 | \( 1 - 12.9T + 67T^{2} \) |
| 71 | \( 1 - 7.32T + 71T^{2} \) |
| 73 | \( 1 + 0.424T + 73T^{2} \) |
| 79 | \( 1 + 6.35T + 79T^{2} \) |
| 83 | \( 1 - 0.737T + 83T^{2} \) |
| 89 | \( 1 + 9.78T + 89T^{2} \) |
| 97 | \( 1 - 0.0337T + 97T^{2} \) |
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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
−9.131989367851976080099219367798, −8.591423831517553615469900490953, −7.63421651649108673533063302769, −7.04540630623396781478839735567, −5.68337284367281557638902707736, −5.21151733988461381514688714167, −4.36847745172376224946419970234, −3.71123637667986172197398487790, −1.88810531120861454427528544026, −0.910298035463927915950174985674,
0.910298035463927915950174985674, 1.88810531120861454427528544026, 3.71123637667986172197398487790, 4.36847745172376224946419970234, 5.21151733988461381514688714167, 5.68337284367281557638902707736, 7.04540630623396781478839735567, 7.63421651649108673533063302769, 8.591423831517553615469900490953, 9.131989367851976080099219367798
