| L(s) = 1 | − 0.246·2-s + 3-s − 1.93·4-s − 0.246·6-s + 1.24·7-s + 0.972·8-s + 9-s + 2.56·11-s − 1.93·12-s + 4.68·13-s − 0.307·14-s + 3.63·16-s − 5.83·17-s − 0.246·18-s + 4.16·19-s + 1.24·21-s − 0.634·22-s − 1.60·23-s + 0.972·24-s − 1.15·26-s + 27-s − 2.41·28-s − 3.21·29-s − 9.19·31-s − 2.84·32-s + 2.56·33-s + 1.44·34-s + ⋯ |
| L(s) = 1 | − 0.174·2-s + 0.577·3-s − 0.969·4-s − 0.100·6-s + 0.471·7-s + 0.343·8-s + 0.333·9-s + 0.774·11-s − 0.559·12-s + 1.29·13-s − 0.0822·14-s + 0.909·16-s − 1.41·17-s − 0.0581·18-s + 0.956·19-s + 0.272·21-s − 0.135·22-s − 0.334·23-s + 0.198·24-s − 0.226·26-s + 0.192·27-s − 0.456·28-s − 0.597·29-s − 1.65·31-s − 0.502·32-s + 0.447·33-s + 0.247·34-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
| \(L(1)\) |
\(\approx\) |
\(1.807612295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807612295\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.246T + 2T^{2} \) |
| 7 | \( 1 - 1.24T + 7T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.68T + 13T^{2} \) |
| 17 | \( 1 + 5.83T + 17T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 + 1.60T + 23T^{2} \) |
| 29 | \( 1 + 3.21T + 29T^{2} \) |
| 31 | \( 1 + 9.19T + 31T^{2} \) |
| 37 | \( 1 - 1.27T + 37T^{2} \) |
| 41 | \( 1 - 8.69T + 41T^{2} \) |
| 43 | \( 1 - 3.88T + 43T^{2} \) |
| 47 | \( 1 - 3.20T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 6.39T + 59T^{2} \) |
| 61 | \( 1 - 3.82T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 2.83T + 83T^{2} \) |
| 89 | \( 1 + 1.38T + 89T^{2} \) |
| 97 | \( 1 + 0.0305T + 97T^{2} \) |
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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
−9.226284162845073892289188240653, −8.568160335062179581180059555069, −7.892165264108486971574147392858, −7.02134672071420653304718933650, −5.98096480932109336322304823135, −5.08927679435100172580253567307, −4.01129684696660647156035085452, −3.68441854093821741487233604506, −2.11023490948991573419923213954, −0.975631266863346111211084477678,
0.975631266863346111211084477678, 2.11023490948991573419923213954, 3.68441854093821741487233604506, 4.01129684696660647156035085452, 5.08927679435100172580253567307, 5.98096480932109336322304823135, 7.02134672071420653304718933650, 7.892165264108486971574147392858, 8.568160335062179581180059555069, 9.226284162845073892289188240653
