| L(s) = 1 | − 5.53·2-s + 3·3-s + 22.6·4-s + 4.34·5-s − 16.6·6-s − 33.2·7-s − 81.1·8-s + 9·9-s − 24.0·10-s + 11·11-s + 67.9·12-s + 13.4·13-s + 184.·14-s + 13.0·15-s + 268.·16-s + 104.·17-s − 49.8·18-s − 93.4·19-s + 98.5·20-s − 99.8·21-s − 60.9·22-s − 18.3·23-s − 243.·24-s − 106.·25-s − 74.6·26-s + 27·27-s − 753.·28-s + ⋯ |
| L(s) = 1 | − 1.95·2-s + 0.577·3-s + 2.83·4-s + 0.388·5-s − 1.13·6-s − 1.79·7-s − 3.58·8-s + 0.333·9-s − 0.761·10-s + 0.301·11-s + 1.63·12-s + 0.287·13-s + 3.51·14-s + 0.224·15-s + 4.18·16-s + 1.49·17-s − 0.652·18-s − 1.12·19-s + 1.10·20-s − 1.03·21-s − 0.590·22-s − 0.166·23-s − 2.07·24-s − 0.848·25-s − 0.563·26-s + 0.192·27-s − 5.08·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 + 5.53T + 8T^{2} \) |
| 5 | \( 1 - 4.34T + 125T^{2} \) |
| 7 | \( 1 + 33.2T + 343T^{2} \) |
| 13 | \( 1 - 13.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 93.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 18.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 103.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 438.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 148.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 39.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 195.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 504.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 518.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 271.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 88.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 224.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 194.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.08e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 85.6T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−8.517778377550238272385553397908, −7.86463184848381864011340357263, −7.06172704185475574204545189095, −6.27126562060786442464136535657, −5.90583571665127957383705830635, −3.74290250079859725196907283609, −2.99301016338190951401775199910, −2.15436424285713044045591362441, −1.05251621076460092696337903279, 0,
1.05251621076460092696337903279, 2.15436424285713044045591362441, 2.99301016338190951401775199910, 3.74290250079859725196907283609, 5.90583571665127957383705830635, 6.27126562060786442464136535657, 7.06172704185475574204545189095, 7.86463184848381864011340357263, 8.517778377550238272385553397908
