| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 5·11-s + 16-s + 2·17-s + 19-s + 20-s + 5·22-s + 23-s − 4·25-s − 4·29-s + 9·31-s − 32-s − 2·34-s + 5·37-s − 38-s − 40-s − 9·41-s − 10·43-s − 5·44-s − 46-s + 6·47-s + 4·50-s − 12·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.50·11-s + 1/4·16-s + 0.485·17-s + 0.229·19-s + 0.223·20-s + 1.06·22-s + 0.208·23-s − 4/5·25-s − 0.742·29-s + 1.61·31-s − 0.176·32-s − 0.342·34-s + 0.821·37-s − 0.162·38-s − 0.158·40-s − 1.40·41-s − 1.52·43-s − 0.753·44-s − 0.147·46-s + 0.875·47-s + 0.565·50-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 13 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.220106110507082378650077005558, −8.010239712591061983505098763067, −7.09420888318909983631668599765, −6.20576320713408039250924060718, −5.47419592206286979467228248399, −4.69043273304129866250258936877, −3.31328950804851415057657782458, −2.53354081754411932509398833785, −1.48581375357083599961757008935, 0,
1.48581375357083599961757008935, 2.53354081754411932509398833785, 3.31328950804851415057657782458, 4.69043273304129866250258936877, 5.47419592206286979467228248399, 6.20576320713408039250924060718, 7.09420888318909983631668599765, 8.010239712591061983505098763067, 8.220106110507082378650077005558
