| L(s) = 1 | − 10.1·2-s − 26.5·3-s + 71.1·4-s + 25·5-s + 269.·6-s + 225.·7-s − 397.·8-s + 461.·9-s − 253.·10-s + 121·11-s − 1.88e3·12-s + 301.·13-s − 2.28e3·14-s − 663.·15-s + 1.75e3·16-s + 2.04e3·17-s − 4.68e3·18-s + 361·19-s + 1.77e3·20-s − 5.98e3·21-s − 1.22e3·22-s − 4.32e3·23-s + 1.05e4·24-s + 625·25-s − 3.06e3·26-s − 5.80e3·27-s + 1.60e4·28-s + ⋯ |
| L(s) = 1 | − 1.79·2-s − 1.70·3-s + 2.22·4-s + 0.447·5-s + 3.05·6-s + 1.73·7-s − 2.19·8-s + 1.89·9-s − 0.802·10-s + 0.301·11-s − 3.78·12-s + 0.494·13-s − 3.12·14-s − 0.761·15-s + 1.71·16-s + 1.71·17-s − 3.40·18-s + 0.229·19-s + 0.993·20-s − 2.96·21-s − 0.541·22-s − 1.70·23-s + 3.73·24-s + 0.200·25-s − 0.888·26-s − 1.53·27-s + 3.86·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7790420773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7790420773\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 10.1T + 32T^{2} \) |
| 3 | \( 1 + 26.5T + 243T^{2} \) |
| 7 | \( 1 - 225.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 301.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 2.04e3T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.32e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.29e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 9.35e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.26e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.01e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.53e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.91e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.75e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.02e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.67e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.27e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.25e5T + 8.58e9T^{2} \) |
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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
−9.360852734749951754674576342090, −8.270735889251003710183711295786, −7.64830131482937931211641644191, −6.90331127918829556587161774412, −5.66674000450584339743156243877, −5.53645600456720606590312182864, −4.04240308226823724897266155939, −1.96530465623887057211574078263, −1.36452986468726712496967186065, −0.62719315628655841889088186893,
0.62719315628655841889088186893, 1.36452986468726712496967186065, 1.96530465623887057211574078263, 4.04240308226823724897266155939, 5.53645600456720606590312182864, 5.66674000450584339743156243877, 6.90331127918829556587161774412, 7.64830131482937931211641644191, 8.270735889251003710183711295786, 9.360852734749951754674576342090
