| L(s) = 1 | + 2·2-s + 4·4-s + 19·7-s + 8·8-s − 12·11-s − 50·13-s + 38·14-s + 16·16-s − 126·17-s + 29·19-s − 24·22-s + 18·23-s − 100·26-s + 76·28-s − 102·29-s − 265·31-s + 32·32-s − 252·34-s − 65·37-s + 58·38-s − 240·41-s + 367·43-s − 48·44-s + 36·46-s − 72·47-s + 18·49-s − 200·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.02·7-s + 0.353·8-s − 0.328·11-s − 1.06·13-s + 0.725·14-s + 1/4·16-s − 1.79·17-s + 0.350·19-s − 0.232·22-s + 0.163·23-s − 0.754·26-s + 0.512·28-s − 0.653·29-s − 1.53·31-s + 0.176·32-s − 1.27·34-s − 0.288·37-s + 0.247·38-s − 0.914·41-s + 1.30·43-s − 0.164·44-s + 0.115·46-s − 0.223·47-s + 0.0524·49-s − 0.533·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 19 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 50 T + p^{3} T^{2} \) |
| 17 | \( 1 + 126 T + p^{3} T^{2} \) |
| 19 | \( 1 - 29 T + p^{3} T^{2} \) |
| 23 | \( 1 - 18 T + p^{3} T^{2} \) |
| 29 | \( 1 + 102 T + p^{3} T^{2} \) |
| 31 | \( 1 + 265 T + p^{3} T^{2} \) |
| 37 | \( 1 + 65 T + p^{3} T^{2} \) |
| 41 | \( 1 + 240 T + p^{3} T^{2} \) |
| 43 | \( 1 - 367 T + p^{3} T^{2} \) |
| 47 | \( 1 + 72 T + p^{3} T^{2} \) |
| 53 | \( 1 + 12 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 102 T + p^{3} T^{2} \) |
| 61 | \( 1 + 103 T + p^{3} T^{2} \) |
| 67 | \( 1 - 52 T + p^{3} T^{2} \) |
| 71 | \( 1 - 582 T + p^{3} T^{2} \) |
| 73 | \( 1 + 65 T + p^{3} T^{2} \) |
| 79 | \( 1 - 173 T + p^{3} T^{2} \) |
| 83 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 89 | \( 1 - 822 T + p^{3} T^{2} \) |
| 97 | \( 1 + 821 T + p^{3} T^{2} \) |
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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
−8.828280272499340759622824533716, −7.79962538829393697607107992617, −7.23268881686899386388625516673, −6.29137809090221721748708593655, −5.14898183850042211124338818850, −4.78013879231852411403794602151, −3.73940642373243556881031663009, −2.47746858820820106691659590480, −1.72960200024121699832248977747, 0,
1.72960200024121699832248977747, 2.47746858820820106691659590480, 3.73940642373243556881031663009, 4.78013879231852411403794602151, 5.14898183850042211124338818850, 6.29137809090221721748708593655, 7.23268881686899386388625516673, 7.79962538829393697607107992617, 8.828280272499340759622824533716
