| L(s) = 1 | + 0.104·2-s − 1.98·4-s + 3.88·7-s − 0.415·8-s + 5.28·11-s − 0.694·13-s + 0.404·14-s + 3.93·16-s + 1.67·17-s + 3.99·19-s + 0.550·22-s + 2.82·23-s − 0.0723·26-s − 7.73·28-s + 8.18·29-s + 31-s + 1.24·32-s + 0.174·34-s + 9.62·37-s + 0.415·38-s − 7.41·41-s − 2.34·43-s − 10.5·44-s + 0.294·46-s − 0.567·47-s + 8.10·49-s + 1.38·52-s + ⋯ |
| L(s) = 1 | + 0.0736·2-s − 0.994·4-s + 1.46·7-s − 0.146·8-s + 1.59·11-s − 0.192·13-s + 0.108·14-s + 0.983·16-s + 0.406·17-s + 0.915·19-s + 0.117·22-s + 0.589·23-s − 0.0141·26-s − 1.46·28-s + 1.52·29-s + 0.179·31-s + 0.219·32-s + 0.0299·34-s + 1.58·37-s + 0.0674·38-s − 1.15·41-s − 0.357·43-s − 1.58·44-s + 0.0434·46-s − 0.0828·47-s + 1.15·49-s + 0.191·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.579044944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.579044944\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 - 0.104T + 2T^{2} \) |
| 7 | \( 1 - 3.88T + 7T^{2} \) |
| 11 | \( 1 - 5.28T + 11T^{2} \) |
| 13 | \( 1 + 0.694T + 13T^{2} \) |
| 17 | \( 1 - 1.67T + 17T^{2} \) |
| 19 | \( 1 - 3.99T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 8.18T + 29T^{2} \) |
| 37 | \( 1 - 9.62T + 37T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 + 2.34T + 43T^{2} \) |
| 47 | \( 1 + 0.567T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 - 4.98T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 8.46T + 73T^{2} \) |
| 79 | \( 1 + 0.437T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 2.97T + 89T^{2} \) |
| 97 | \( 1 + 9.79T + 97T^{2} \) |
| 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
−8.015180807482821771969138979615, −7.41154978394241045256568716139, −6.52271711870150466266290940335, −5.69474371715935321759426983605, −4.92326630857348648199782700839, −4.49632314182035845465952150670, −3.76806886736072523694138873220, −2.83853600859572577742786657284, −1.45443528559924528727541373108, −0.965575309820198030017839527196,
0.965575309820198030017839527196, 1.45443528559924528727541373108, 2.83853600859572577742786657284, 3.76806886736072523694138873220, 4.49632314182035845465952150670, 4.92326630857348648199782700839, 5.69474371715935321759426983605, 6.52271711870150466266290940335, 7.41154978394241045256568716139, 8.015180807482821771969138979615
