| L(s) = 1 | + 5.11·2-s − 8.22·3-s + 18.1·4-s − 5·5-s − 42.0·6-s − 10.3·7-s + 51.8·8-s + 40.6·9-s − 25.5·10-s + 2.41·11-s − 149.·12-s − 13·13-s − 52.8·14-s + 41.1·15-s + 120.·16-s − 23.6·17-s + 208.·18-s + 148.·19-s − 90.7·20-s + 85.0·21-s + 12.3·22-s − 48.4·23-s − 426.·24-s + 25·25-s − 66.4·26-s − 112.·27-s − 187.·28-s + ⋯ |
| L(s) = 1 | + 1.80·2-s − 1.58·3-s + 2.26·4-s − 0.447·5-s − 2.86·6-s − 0.558·7-s + 2.29·8-s + 1.50·9-s − 0.808·10-s + 0.0661·11-s − 3.59·12-s − 0.277·13-s − 1.00·14-s + 0.708·15-s + 1.87·16-s − 0.337·17-s + 2.72·18-s + 1.79·19-s − 1.01·20-s + 0.883·21-s + 0.119·22-s − 0.439·23-s − 3.63·24-s + 0.200·25-s − 0.501·26-s − 0.803·27-s − 1.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 5T \) |
| 13 | \( 1 + 13T \) |
| 31 | \( 1 - 31T \) |
| good | 2 | \( 1 - 5.11T + 8T^{2} \) |
| 3 | \( 1 + 8.22T + 27T^{2} \) |
| 7 | \( 1 + 10.3T + 343T^{2} \) |
| 11 | \( 1 - 2.41T + 1.33e3T^{2} \) |
| 17 | \( 1 + 23.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 148.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 48.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 10.3T + 2.43e4T^{2} \) |
| 37 | \( 1 + 47.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 466.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 542.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 241.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 41.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 291.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 163.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 636.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 538.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 603.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 221.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.03e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 753.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 189.T + 9.12e5T^{2} \) |
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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
−7.937272368682863465884182450120, −7.01753916987006626929382184764, −6.57865037868301664348407325916, −5.76863925963013143110020294619, −5.19989888575987641422105745275, −4.53846194553031230111700950471, −3.67066492723671807516200742330, −2.80048742552271255469022060556, −1.33616139759585960744521614157, 0,
1.33616139759585960744521614157, 2.80048742552271255469022060556, 3.67066492723671807516200742330, 4.53846194553031230111700950471, 5.19989888575987641422105745275, 5.76863925963013143110020294619, 6.57865037868301664348407325916, 7.01753916987006626929382184764, 7.937272368682863465884182450120
