| L(s) = 1 | − 2-s − 2.42·3-s + 4-s + 2.60·5-s + 2.42·6-s + 7-s − 8-s + 2.88·9-s − 2.60·10-s − 1.54·11-s − 2.42·12-s + 3.31·13-s − 14-s − 6.31·15-s + 16-s + 3.53·17-s − 2.88·18-s + 7.64·19-s + 2.60·20-s − 2.42·21-s + 1.54·22-s + 3.54·23-s + 2.42·24-s + 1.78·25-s − 3.31·26-s + 0.287·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.40·3-s + 0.5·4-s + 1.16·5-s + 0.990·6-s + 0.377·7-s − 0.353·8-s + 0.960·9-s − 0.823·10-s − 0.465·11-s − 0.700·12-s + 0.918·13-s − 0.267·14-s − 1.63·15-s + 0.250·16-s + 0.858·17-s − 0.679·18-s + 1.75·19-s + 0.582·20-s − 0.529·21-s + 0.329·22-s + 0.739·23-s + 0.495·24-s + 0.357·25-s − 0.649·26-s + 0.0553·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.390332626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.390332626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 2.42T + 3T^{2} \) |
| 5 | \( 1 - 2.60T + 5T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 - 3.31T + 13T^{2} \) |
| 17 | \( 1 - 3.53T + 17T^{2} \) |
| 19 | \( 1 - 7.64T + 19T^{2} \) |
| 23 | \( 1 - 3.54T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 - 4.95T + 31T^{2} \) |
| 37 | \( 1 + 9.92T + 37T^{2} \) |
| 41 | \( 1 - 3.22T + 41T^{2} \) |
| 43 | \( 1 - 1.21T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 5.48T + 53T^{2} \) |
| 59 | \( 1 - 5.29T + 59T^{2} \) |
| 61 | \( 1 + 6.17T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 2.55T + 71T^{2} \) |
| 73 | \( 1 - 6.08T + 73T^{2} \) |
| 79 | \( 1 - 8.36T + 79T^{2} \) |
| 83 | \( 1 + 3.64T + 83T^{2} \) |
| 89 | \( 1 - 4.60T + 89T^{2} \) |
| 97 | \( 1 + 6.96T + 97T^{2} \) |
| 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
−8.049318408945938297443037312107, −7.28649083061667732644546321692, −6.53020496895870809298174170748, −5.95629692945564502959363197089, −5.31731465538677444091367159909, −4.97659008022381150084637888260, −3.52789165206068614779288725495, −2.56145986920209276892792620212, −1.36025368359856707876247059947, −0.864330722748682617707486968052,
0.864330722748682617707486968052, 1.36025368359856707876247059947, 2.56145986920209276892792620212, 3.52789165206068614779288725495, 4.97659008022381150084637888260, 5.31731465538677444091367159909, 5.95629692945564502959363197089, 6.53020496895870809298174170748, 7.28649083061667732644546321692, 8.049318408945938297443037312107
