| L(s) = 1 | + 2.77·2-s − 2.47·3-s + 5.69·4-s + 3.82·5-s − 6.85·6-s + 0.873·7-s + 10.2·8-s + 3.10·9-s + 10.6·10-s − 4.12·11-s − 14.0·12-s + 2.04·13-s + 2.42·14-s − 9.45·15-s + 17.0·16-s − 7.61·17-s + 8.60·18-s − 7.21·19-s + 21.7·20-s − 2.15·21-s − 11.4·22-s − 2.09·23-s − 25.2·24-s + 9.63·25-s + 5.67·26-s − 0.254·27-s + 4.96·28-s + ⋯ |
| L(s) = 1 | + 1.96·2-s − 1.42·3-s + 2.84·4-s + 1.71·5-s − 2.79·6-s + 0.329·7-s + 3.62·8-s + 1.03·9-s + 3.35·10-s − 1.24·11-s − 4.05·12-s + 0.567·13-s + 0.647·14-s − 2.44·15-s + 4.25·16-s − 1.84·17-s + 2.02·18-s − 1.65·19-s + 4.86·20-s − 0.470·21-s − 2.43·22-s − 0.436·23-s − 5.16·24-s + 1.92·25-s + 1.11·26-s − 0.0489·27-s + 0.939·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.211287695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.211287695\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 787 | \( 1 - T \) |
| good | 2 | \( 1 - 2.77T + 2T^{2} \) |
| 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 7 | \( 1 - 0.873T + 7T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 17 | \( 1 + 7.61T + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + 2.09T + 23T^{2} \) |
| 29 | \( 1 - 5.15T + 29T^{2} \) |
| 31 | \( 1 - 3.13T + 31T^{2} \) |
| 37 | \( 1 - 2.61T + 37T^{2} \) |
| 41 | \( 1 - 1.03T + 41T^{2} \) |
| 43 | \( 1 + 8.88T + 43T^{2} \) |
| 47 | \( 1 - 2.11T + 47T^{2} \) |
| 53 | \( 1 - 4.20T + 53T^{2} \) |
| 59 | \( 1 + 1.46T + 59T^{2} \) |
| 61 | \( 1 + 4.21T + 61T^{2} \) |
| 67 | \( 1 + 9.57T + 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 8.12T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 1.94T + 97T^{2} \) |
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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
−10.70535784276720520839159107763, −10.16566379155543412621236201634, −8.406313445926751114776016966500, −6.82591706245375773834787504062, −6.27778198398925695316893412585, −5.85792461769750099594999473799, −4.91468296858149692809589809304, −4.50255757737248135279072711636, −2.63681774206852460188681512234, −1.83449352001252529273382618366,
1.83449352001252529273382618366, 2.63681774206852460188681512234, 4.50255757737248135279072711636, 4.91468296858149692809589809304, 5.85792461769750099594999473799, 6.27778198398925695316893412585, 6.82591706245375773834787504062, 8.406313445926751114776016966500, 10.16566379155543412621236201634, 10.70535784276720520839159107763
