| L(s) = 1 | − 0.183·2-s + 3.29·3-s − 1.96·4-s − 0.605·6-s − 3.65·7-s + 0.728·8-s + 7.85·9-s − 0.912·11-s − 6.47·12-s + 1.27·13-s + 0.670·14-s + 3.79·16-s − 1.65·17-s − 1.44·18-s + 3.74·19-s − 12.0·21-s + 0.167·22-s + 0.248·23-s + 2.40·24-s − 0.234·26-s + 15.9·27-s + 7.17·28-s + 1.93·29-s − 5.44·31-s − 2.15·32-s − 3.00·33-s + 0.304·34-s + ⋯ |
| L(s) = 1 | − 0.129·2-s + 1.90·3-s − 0.983·4-s − 0.247·6-s − 1.37·7-s + 0.257·8-s + 2.61·9-s − 0.275·11-s − 1.87·12-s + 0.354·13-s + 0.179·14-s + 0.949·16-s − 0.402·17-s − 0.340·18-s + 0.859·19-s − 2.62·21-s + 0.0357·22-s + 0.0518·23-s + 0.489·24-s − 0.0460·26-s + 3.07·27-s + 1.35·28-s + 0.359·29-s − 0.977·31-s − 0.380·32-s − 0.523·33-s + 0.0522·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.540365367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.540365367\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 151 | \( 1 - T \) |
| good | 2 | \( 1 + 0.183T + 2T^{2} \) |
| 3 | \( 1 - 3.29T + 3T^{2} \) |
| 7 | \( 1 + 3.65T + 7T^{2} \) |
| 11 | \( 1 + 0.912T + 11T^{2} \) |
| 13 | \( 1 - 1.27T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 - 3.74T + 19T^{2} \) |
| 23 | \( 1 - 0.248T + 23T^{2} \) |
| 29 | \( 1 - 1.93T + 29T^{2} \) |
| 31 | \( 1 + 5.44T + 31T^{2} \) |
| 37 | \( 1 - 4.41T + 37T^{2} \) |
| 41 | \( 1 - 8.64T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 0.862T + 59T^{2} \) |
| 61 | \( 1 + 2.48T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 - 8.78T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 7.06T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 0.860T + 97T^{2} \) |
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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
−8.670150305003460103521940106845, −7.928811691938055434382136342434, −7.32560770766902658988424432029, −6.50465324577559252928916416718, −5.40792915681625406136898730970, −4.33425045681947844778346266718, −3.67454188351768651807841066514, −3.13294950223027443255684475425, −2.25250289729106098915770881242, −0.879233845169050198469123363069,
0.879233845169050198469123363069, 2.25250289729106098915770881242, 3.13294950223027443255684475425, 3.67454188351768651807841066514, 4.33425045681947844778346266718, 5.40792915681625406136898730970, 6.50465324577559252928916416718, 7.32560770766902658988424432029, 7.928811691938055434382136342434, 8.670150305003460103521940106845
