L(s) = 1 | + 7.46·2-s − 24.9·3-s + 23.7·4-s − 25·5-s − 186.·6-s − 45.5·7-s − 61.5·8-s + 377.·9-s − 186.·10-s + 121·11-s − 591.·12-s − 988.·13-s − 339.·14-s + 622.·15-s − 1.21e3·16-s − 633.·17-s + 2.81e3·18-s − 361·19-s − 593.·20-s + 1.13e3·21-s + 903.·22-s − 3.18e3·23-s + 1.53e3·24-s + 625·25-s − 7.37e3·26-s − 3.35e3·27-s − 1.08e3·28-s + ⋯ |
L(s) = 1 | + 1.32·2-s − 1.59·3-s + 0.742·4-s − 0.447·5-s − 2.10·6-s − 0.351·7-s − 0.339·8-s + 1.55·9-s − 0.590·10-s + 0.301·11-s − 1.18·12-s − 1.62·13-s − 0.463·14-s + 0.714·15-s − 1.19·16-s − 0.531·17-s + 2.05·18-s − 0.229·19-s − 0.332·20-s + 0.561·21-s + 0.398·22-s − 1.25·23-s + 0.543·24-s + 0.200·25-s − 2.14·26-s − 0.884·27-s − 0.260·28-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
\(L(3)\) |
\(\approx\) |
\(0.01928124904\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01928124904\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 - 7.46T + 32T^{2} \) |
| 3 | \( 1 + 24.9T + 243T^{2} \) |
| 7 | \( 1 + 45.5T + 1.68e4T^{2} \) |
| 13 | \( 1 + 988.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 633.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.18e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.94e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 223.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.76e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.15e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.39e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.56e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.37e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.81e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.28e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.74e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.40e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.67e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.16e5T + 8.58e9T^{2} \) |
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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
−9.514015328925518202630115765467, −8.134217979628084158944264282941, −6.84903015094715002283271213788, −6.56172354326666856491436094114, −5.53710038655686607276373450992, −4.91075107812877068323567600874, −4.28532790096341322637879448481, −3.25994604836089932457172044478, −1.91543587799810754147567336947, −0.05012531932897915162140909335,
0.05012531932897915162140909335, 1.91543587799810754147567336947, 3.25994604836089932457172044478, 4.28532790096341322637879448481, 4.91075107812877068323567600874, 5.53710038655686607276373450992, 6.56172354326666856491436094114, 6.84903015094715002283271213788, 8.134217979628084158944264282941, 9.514015328925518202630115765467