L(s) = 1 | − 2·4-s − 5-s − 7-s + 6·11-s + 2·13-s + 4·16-s − 6·17-s − 4·19-s + 2·20-s − 6·23-s + 25-s + 2·28-s + 5·31-s + 35-s + 5·37-s − 6·41-s − 7·43-s − 12·44-s − 6·47-s − 6·49-s − 4·52-s − 12·53-s − 6·55-s − 6·59-s − 61-s − 8·64-s − 2·65-s + ⋯ |
L(s) = 1 | − 4-s − 0.447·5-s − 0.377·7-s + 1.80·11-s + 0.554·13-s + 16-s − 1.45·17-s − 0.917·19-s + 0.447·20-s − 1.25·23-s + 1/5·25-s + 0.377·28-s + 0.898·31-s + 0.169·35-s + 0.821·37-s − 0.937·41-s − 1.06·43-s − 1.80·44-s − 0.875·47-s − 6/7·49-s − 0.554·52-s − 1.64·53-s − 0.809·55-s − 0.781·59-s − 0.128·61-s − 64-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.233326408332339673171918010705, −8.614594303594205062766360400744, −7.949362651083096423973227490088, −6.43604550557634588601712739640, −6.35760940343674826670067311851, −4.69311199846157257103632609233, −4.16522638795181266302474254737, −3.35587289212952129896046717701, −1.60414437646076078479272454891, 0,
1.60414437646076078479272454891, 3.35587289212952129896046717701, 4.16522638795181266302474254737, 4.69311199846157257103632609233, 6.35760940343674826670067311851, 6.43604550557634588601712739640, 7.949362651083096423973227490088, 8.614594303594205062766360400744, 9.233326408332339673171918010705