| L(s) = 1 | − 2-s + 4-s − 5-s − 3·7-s − 8-s + 10-s + 5·11-s + 3·14-s + 16-s + 2·17-s − 19-s − 20-s − 5·22-s − 7·23-s + 25-s − 3·28-s − 4·29-s − 31-s − 32-s − 2·34-s + 3·35-s − 8·37-s + 38-s + 40-s + 12·41-s − 43-s + 5·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.13·7-s − 0.353·8-s + 0.316·10-s + 1.50·11-s + 0.801·14-s + 1/4·16-s + 0.485·17-s − 0.229·19-s − 0.223·20-s − 1.06·22-s − 1.45·23-s + 1/5·25-s − 0.566·28-s − 0.742·29-s − 0.179·31-s − 0.176·32-s − 0.342·34-s + 0.507·35-s − 1.31·37-s + 0.162·38-s + 0.158·40-s + 1.87·41-s − 0.152·43-s + 0.753·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 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 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−8.623563768761589495034153164735, −7.60781151070551273888285334452, −7.03195079752613144055272012834, −6.24973845082477982686825929898, −5.66665296857389448622094723191, −4.09018548716333501193536086062, −3.69297455811894264063722774036, −2.55175262249437549314083320708, −1.31330276315476452677315623294, 0,
1.31330276315476452677315623294, 2.55175262249437549314083320708, 3.69297455811894264063722774036, 4.09018548716333501193536086062, 5.66665296857389448622094723191, 6.24973845082477982686825929898, 7.03195079752613144055272012834, 7.60781151070551273888285334452, 8.623563768761589495034153164735
