| L(s) = 1 | + 1.83·2-s + 0.860·3-s + 1.36·4-s − 3.15·5-s + 1.57·6-s + 2.30·7-s − 1.17·8-s − 2.26·9-s − 5.79·10-s + 4.75·11-s + 1.17·12-s − 6.46·13-s + 4.22·14-s − 2.71·15-s − 4.86·16-s + 6.20·17-s − 4.14·18-s − 1.72·19-s − 4.29·20-s + 1.98·21-s + 8.71·22-s + 7.50·23-s − 1.00·24-s + 4.97·25-s − 11.8·26-s − 4.52·27-s + 3.13·28-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.496·3-s + 0.680·4-s − 1.41·5-s + 0.643·6-s + 0.871·7-s − 0.414·8-s − 0.753·9-s − 1.83·10-s + 1.43·11-s + 0.337·12-s − 1.79·13-s + 1.12·14-s − 0.701·15-s − 1.21·16-s + 1.50·17-s − 0.976·18-s − 0.396·19-s − 0.961·20-s + 0.432·21-s + 1.85·22-s + 1.56·23-s − 0.205·24-s + 0.995·25-s − 2.32·26-s − 0.870·27-s + 0.592·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.713743317\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.713743317\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 103 | \( 1 - T \) |
| good | 2 | \( 1 - 1.83T + 2T^{2} \) |
| 3 | \( 1 - 0.860T + 3T^{2} \) |
| 5 | \( 1 + 3.15T + 5T^{2} \) |
| 7 | \( 1 - 2.30T + 7T^{2} \) |
| 11 | \( 1 - 4.75T + 11T^{2} \) |
| 13 | \( 1 + 6.46T + 13T^{2} \) |
| 17 | \( 1 - 6.20T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 - 7.50T + 23T^{2} \) |
| 29 | \( 1 - 0.513T + 29T^{2} \) |
| 31 | \( 1 + 1.19T + 31T^{2} \) |
| 37 | \( 1 + 1.39T + 37T^{2} \) |
| 41 | \( 1 - 6.48T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 + 6.86T + 47T^{2} \) |
| 53 | \( 1 + 1.24T + 53T^{2} \) |
| 59 | \( 1 - 9.79T + 59T^{2} \) |
| 61 | \( 1 + 1.04T + 61T^{2} \) |
| 67 | \( 1 - 4.24T + 67T^{2} \) |
| 71 | \( 1 + 5.53T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 - 9.28T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 + 18.0T + 89T^{2} \) |
| 97 | \( 1 - 4.11T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.31018349187541779151783005312, −12.63955522412295395783463495669, −11.87624188507729901316211888018, −11.33768718037512105669934081016, −9.315996717047749507620909322653, −8.149295837875702553198146065356, −7.06729883349295601297497109887, −5.27020083146434998669067164559, −4.20699012267397293073228377027, −3.06311476163509552982120840373,
3.06311476163509552982120840373, 4.20699012267397293073228377027, 5.27020083146434998669067164559, 7.06729883349295601297497109887, 8.149295837875702553198146065356, 9.315996717047749507620909322653, 11.33768718037512105669934081016, 11.87624188507729901316211888018, 12.63955522412295395783463495669, 14.31018349187541779151783005312
