| L(s) = 1 | − 2·2-s + 2·4-s + 7-s + 5·11-s + 2·13-s − 2·14-s − 4·16-s − 10·22-s + 2·23-s − 4·26-s + 2·28-s − 6·29-s − 4·31-s + 8·32-s + 37-s + 9·41-s − 2·43-s + 10·44-s − 4·46-s − 9·47-s − 6·49-s + 4·52-s + 53-s + 12·58-s − 8·59-s − 8·61-s + 8·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.377·7-s + 1.50·11-s + 0.554·13-s − 0.534·14-s − 16-s − 2.13·22-s + 0.417·23-s − 0.784·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 1.41·32-s + 0.164·37-s + 1.40·41-s − 0.304·43-s + 1.50·44-s − 0.589·46-s − 1.31·47-s − 6/7·49-s + 0.554·52-s + 0.137·53-s + 1.57·58-s − 1.04·59-s − 1.02·61-s + 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.65121136122522161197318054345, −6.94384664150198731468349379760, −6.36962640367681528657535462564, −5.55465832553444053493877306394, −4.50549554166763456403503901204, −3.91879626410547859574461452178, −2.89439405381897961095418638007, −1.61293502722827766917172111896, −1.35033591033027974757425948725, 0,
1.35033591033027974757425948725, 1.61293502722827766917172111896, 2.89439405381897961095418638007, 3.91879626410547859574461452178, 4.50549554166763456403503901204, 5.55465832553444053493877306394, 6.36962640367681528657535462564, 6.94384664150198731468349379760, 7.65121136122522161197318054345
