| L(s) = 1 | + 2-s − 3.11·3-s + 4-s + 2.93·5-s − 3.11·6-s + 4.25·7-s + 8-s + 6.67·9-s + 2.93·10-s − 11-s − 3.11·12-s + 4.02·13-s + 4.25·14-s − 9.11·15-s + 16-s − 3.10·17-s + 6.67·18-s + 2.93·20-s − 13.2·21-s − 22-s + 8.63·23-s − 3.11·24-s + 3.59·25-s + 4.02·26-s − 11.4·27-s + 4.25·28-s + 1.90·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.79·3-s + 0.5·4-s + 1.31·5-s − 1.27·6-s + 1.60·7-s + 0.353·8-s + 2.22·9-s + 0.926·10-s − 0.301·11-s − 0.898·12-s + 1.11·13-s + 1.13·14-s − 2.35·15-s + 0.250·16-s − 0.753·17-s + 1.57·18-s + 0.655·20-s − 2.89·21-s − 0.213·22-s + 1.79·23-s − 0.635·24-s + 0.718·25-s + 0.789·26-s − 2.20·27-s + 0.804·28-s + 0.354·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.345089456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.345089456\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 3.11T + 3T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 + 3.10T + 17T^{2} \) |
| 23 | \( 1 - 8.63T + 23T^{2} \) |
| 29 | \( 1 - 1.90T + 29T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 + 6.62T + 37T^{2} \) |
| 41 | \( 1 - 4.69T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 - 5.87T + 47T^{2} \) |
| 53 | \( 1 - 0.147T + 53T^{2} \) |
| 59 | \( 1 - 4.30T + 59T^{2} \) |
| 61 | \( 1 + 7.53T + 61T^{2} \) |
| 67 | \( 1 - 2.17T + 67T^{2} \) |
| 71 | \( 1 - 13.7T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 0.921T + 79T^{2} \) |
| 83 | \( 1 - 0.684T + 83T^{2} \) |
| 89 | \( 1 + 13.4T + 89T^{2} \) |
| 97 | \( 1 - 16.7T + 97T^{2} \) |
| 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
−7.45404867859389590065929304129, −6.88592967640019859540239210658, −6.12924539477425536558930468915, −5.72388173496940269041875899274, −5.01692057585122419999060079488, −4.80782657666912414391825377325, −3.87159687893510848468167550197, −2.45516351770649981697556762127, −1.58150768126396126877069543429, −1.01273170111468682354955230828,
1.01273170111468682354955230828, 1.58150768126396126877069543429, 2.45516351770649981697556762127, 3.87159687893510848468167550197, 4.80782657666912414391825377325, 5.01692057585122419999060079488, 5.72388173496940269041875899274, 6.12924539477425536558930468915, 6.88592967640019859540239210658, 7.45404867859389590065929304129
