| L(s) = 1 | − 2-s + 4-s − 5-s − 3·7-s − 8-s + 10-s − 11-s − 13-s + 3·14-s + 16-s + 7·17-s − 19-s − 20-s + 22-s + 3·23-s + 25-s + 26-s − 3·28-s − 4·31-s − 32-s − 7·34-s + 3·35-s − 37-s + 38-s + 40-s + 6·41-s + 4·43-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 − 0.301·11-s − 0.277·13-s + 0.801·14-s + 1/4·16-s + 1.69·17-s − 0.229·19-s − 0.223·20-s + 0.213·22-s + 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.566·28-s − 0.718·31-s − 0.176·32-s − 1.20·34-s + 0.507·35-s − 0.164·37-s + 0.162·38-s + 0.158·40-s + 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 7 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.266443917246252976510048296319, −7.43317320543162036533592286864, −7.08534861525043097568933914515, −6.01998294555919825160796329515, −5.46223340748589549036169107920, −4.21410538431347344938634687551, −3.28218319754867075582483662602, −2.66273414150940285121225074429, −1.20174460971488557683221711507, 0,
1.20174460971488557683221711507, 2.66273414150940285121225074429, 3.28218319754867075582483662602, 4.21410538431347344938634687551, 5.46223340748589549036169107920, 6.01998294555919825160796329515, 7.08534861525043097568933914515, 7.43317320543162036533592286864, 8.266443917246252976510048296319
