| L(s) = 1 | + 2-s − 0.0978·3-s + 4-s + 2.81·5-s − 0.0978·6-s + 3.51·7-s + 8-s − 2.99·9-s + 2.81·10-s + 2.08·11-s − 0.0978·12-s + 4.21·13-s + 3.51·14-s − 0.274·15-s + 16-s − 7.72·17-s − 2.99·18-s + 19-s + 2.81·20-s − 0.343·21-s + 2.08·22-s − 7.38·23-s − 0.0978·24-s + 2.89·25-s + 4.21·26-s + 0.586·27-s + 3.51·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.0564·3-s + 0.5·4-s + 1.25·5-s − 0.0399·6-s + 1.32·7-s + 0.353·8-s − 0.996·9-s + 0.888·10-s + 0.628·11-s − 0.0282·12-s + 1.16·13-s + 0.938·14-s − 0.0709·15-s + 0.250·16-s − 1.87·17-s − 0.704·18-s + 0.229·19-s + 0.628·20-s − 0.0749·21-s + 0.444·22-s − 1.54·23-s − 0.0199·24-s + 0.579·25-s + 0.826·26-s + 0.112·27-s + 0.663·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.039650159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.039650159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 0.0978T + 3T^{2} \) |
| 5 | \( 1 - 2.81T + 5T^{2} \) |
| 7 | \( 1 - 3.51T + 7T^{2} \) |
| 11 | \( 1 - 2.08T + 11T^{2} \) |
| 13 | \( 1 - 4.21T + 13T^{2} \) |
| 17 | \( 1 + 7.72T + 17T^{2} \) |
| 23 | \( 1 + 7.38T + 23T^{2} \) |
| 29 | \( 1 - 7.64T + 29T^{2} \) |
| 31 | \( 1 - 0.635T + 31T^{2} \) |
| 37 | \( 1 - 4.20T + 37T^{2} \) |
| 41 | \( 1 - 12.7T + 41T^{2} \) |
| 43 | \( 1 + 8.64T + 43T^{2} \) |
| 47 | \( 1 + 7.96T + 47T^{2} \) |
| 53 | \( 1 - 7.73T + 53T^{2} \) |
| 59 | \( 1 - 13.4T + 59T^{2} \) |
| 61 | \( 1 + 9.51T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 3.11T + 71T^{2} \) |
| 73 | \( 1 - 4.99T + 73T^{2} \) |
| 79 | \( 1 - 5.74T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 - 6.24T + 89T^{2} \) |
| 97 | \( 1 + 5.81T + 97T^{2} \) |
| 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
−8.067953166078442258779486831027, −6.74089514318663725082784491394, −6.28598764991143278539647413238, −5.82270428761836166947338179981, −5.04196278942021967244302951541, −4.41600163402232278843033266193, −3.64521306340835943991048110031, −2.39542458664686714248866383131, −2.06502936251293761948876467382, −1.05737458791765307604315751528,
1.05737458791765307604315751528, 2.06502936251293761948876467382, 2.39542458664686714248866383131, 3.64521306340835943991048110031, 4.41600163402232278843033266193, 5.04196278942021967244302951541, 5.82270428761836166947338179981, 6.28598764991143278539647413238, 6.74089514318663725082784491394, 8.067953166078442258779486831027
