| L(s) = 1 | − 2.28·2-s + 3.22·4-s − 2.86·5-s + 3.60·7-s − 2.79·8-s + 6.53·10-s − 11-s − 3.97·13-s − 8.22·14-s − 0.0663·16-s − 2.89·17-s + 3.91·19-s − 9.21·20-s + 2.28·22-s + 4.02·23-s + 3.18·25-s + 9.07·26-s + 11.5·28-s − 3.89·29-s − 0.470·31-s + 5.73·32-s + 6.62·34-s − 10.2·35-s − 1.32·37-s − 8.94·38-s + 7.98·40-s + 2.18·41-s + ⋯ |
| L(s) = 1 | − 1.61·2-s + 1.61·4-s − 1.27·5-s + 1.36·7-s − 0.986·8-s + 2.06·10-s − 0.301·11-s − 1.10·13-s − 2.19·14-s − 0.0165·16-s − 0.702·17-s + 0.897·19-s − 2.06·20-s + 0.487·22-s + 0.839·23-s + 0.636·25-s + 1.78·26-s + 2.19·28-s − 0.722·29-s − 0.0845·31-s + 1.01·32-s + 1.13·34-s − 1.74·35-s − 0.217·37-s − 1.45·38-s + 1.26·40-s + 0.340·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 2.28T + 2T^{2} \) |
| 5 | \( 1 + 2.86T + 5T^{2} \) |
| 7 | \( 1 - 3.60T + 7T^{2} \) |
| 13 | \( 1 + 3.97T + 13T^{2} \) |
| 17 | \( 1 + 2.89T + 17T^{2} \) |
| 19 | \( 1 - 3.91T + 19T^{2} \) |
| 23 | \( 1 - 4.02T + 23T^{2} \) |
| 29 | \( 1 + 3.89T + 29T^{2} \) |
| 31 | \( 1 + 0.470T + 31T^{2} \) |
| 37 | \( 1 + 1.32T + 37T^{2} \) |
| 41 | \( 1 - 2.18T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 6.67T + 47T^{2} \) |
| 53 | \( 1 - 2.86T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 67 | \( 1 - 6.63T + 67T^{2} \) |
| 71 | \( 1 - 2.50T + 71T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 + 1.93T + 79T^{2} \) |
| 83 | \( 1 + 9.23T + 83T^{2} \) |
| 89 | \( 1 + 6.85T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.81487691093837655674674524204, −7.34333406237250913480840110648, −6.94681787991924057720079773595, −5.55697340600198457309081592405, −4.76645915726043724831174393808, −4.13495562339209526268328806047, −2.88818862101667318278340603387, −2.03328329838141412288412588247, −1.02188773880541790902198891837, 0,
1.02188773880541790902198891837, 2.03328329838141412288412588247, 2.88818862101667318278340603387, 4.13495562339209526268328806047, 4.76645915726043724831174393808, 5.55697340600198457309081592405, 6.94681787991924057720079773595, 7.34333406237250913480840110648, 7.81487691093837655674674524204
