| L(s) = 1 | − 0.271·2-s − 0.319·3-s − 1.92·4-s + 5-s + 0.0867·6-s − 3.38·7-s + 1.06·8-s − 2.89·9-s − 0.271·10-s − 1.75·11-s + 0.615·12-s + 0.917·14-s − 0.319·15-s + 3.56·16-s + 1.95·17-s + 0.786·18-s + 7.13·19-s − 1.92·20-s + 1.08·21-s + 0.475·22-s + 7.61·23-s − 0.340·24-s + 25-s + 1.88·27-s + 6.51·28-s + 3.98·29-s + 0.0867·30-s + ⋯ |
| L(s) = 1 | − 0.191·2-s − 0.184·3-s − 0.963·4-s + 0.447·5-s + 0.0354·6-s − 1.27·7-s + 0.376·8-s − 0.965·9-s − 0.0858·10-s − 0.528·11-s + 0.177·12-s + 0.245·14-s − 0.0825·15-s + 0.890·16-s + 0.473·17-s + 0.185·18-s + 1.63·19-s − 0.430·20-s + 0.235·21-s + 0.101·22-s + 1.58·23-s − 0.0695·24-s + 0.200·25-s + 0.362·27-s + 1.23·28-s + 0.739·29-s + 0.0158·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8196342881\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8196342881\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.271T + 2T^{2} \) |
| 3 | \( 1 + 0.319T + 3T^{2} \) |
| 7 | \( 1 + 3.38T + 7T^{2} \) |
| 11 | \( 1 + 1.75T + 11T^{2} \) |
| 17 | \( 1 - 1.95T + 17T^{2} \) |
| 19 | \( 1 - 7.13T + 19T^{2} \) |
| 23 | \( 1 - 7.61T + 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 + 4.86T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 0.911T + 41T^{2} \) |
| 43 | \( 1 - 4.58T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 + 3.82T + 59T^{2} \) |
| 61 | \( 1 - 7.98T + 61T^{2} \) |
| 67 | \( 1 + 0.472T + 67T^{2} \) |
| 71 | \( 1 - 5.50T + 71T^{2} \) |
| 73 | \( 1 + 2.93T + 73T^{2} \) |
| 79 | \( 1 - 4.09T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 - 3.85T + 89T^{2} \) |
| 97 | \( 1 - 6.95T + 97T^{2} \) |
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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
−10.11311024711985584687483916484, −9.223639086704692898638715936560, −8.868008881014995982566956515239, −7.67192026637481382685444058959, −6.74396191975879274920942748418, −5.49346724066332969311307912503, −5.24729441777454047094827014620, −3.59848959900980891500934654056, −2.85336288390055707206765266853, −0.75689967584385334958717938857,
0.75689967584385334958717938857, 2.85336288390055707206765266853, 3.59848959900980891500934654056, 5.24729441777454047094827014620, 5.49346724066332969311307912503, 6.74396191975879274920942748418, 7.67192026637481382685444058959, 8.868008881014995982566956515239, 9.223639086704692898638715936560, 10.11311024711985584687483916484
