| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 6·11-s + 2·13-s + 14-s + 16-s + 19-s + 6·22-s − 6·23-s − 5·25-s + 2·26-s + 28-s + 6·29-s + 8·31-s + 32-s − 10·37-s + 38-s − 6·41-s − 4·43-s + 6·44-s − 6·46-s + 6·47-s + 49-s − 5·50-s + 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.80·11-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.229·19-s + 1.27·22-s − 1.25·23-s − 25-s + 0.392·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s + 0.176·32-s − 1.64·37-s + 0.162·38-s − 0.937·41-s − 0.609·43-s + 0.904·44-s − 0.884·46-s + 0.875·47-s + 1/7·49-s − 0.707·50-s + 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.421819845\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.421819845\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−8.765582553646625046364090401588, −8.307304321723518830188404349040, −7.21905434487687976551028101741, −6.49153351787333795376314013168, −5.93407852478313956868198078366, −4.91274849590239228277337676397, −4.03629244778711706613734856845, −3.51976824410808013707609172995, −2.15567949814509121272859484390, −1.18917455220197181454883757861,
1.18917455220197181454883757861, 2.15567949814509121272859484390, 3.51976824410808013707609172995, 4.03629244778711706613734856845, 4.91274849590239228277337676397, 5.93407852478313956868198078366, 6.49153351787333795376314013168, 7.21905434487687976551028101741, 8.307304321723518830188404349040, 8.765582553646625046364090401588
