| L(s) = 1 | − 5-s + 0.828·7-s + 2·11-s − 6.24·13-s − 0.828·17-s − 19-s + 6·23-s + 25-s + 6.48·29-s − 6.82·31-s − 0.828·35-s − 1.75·37-s − 3.65·41-s + 4.82·43-s + 4.82·47-s − 6.31·49-s − 9.07·53-s − 2·55-s − 13.6·59-s − 13.6·61-s + 6.24·65-s − 3.41·67-s − 5.17·71-s − 2.48·73-s + 1.65·77-s + 1.65·79-s + 13.3·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.313·7-s + 0.603·11-s − 1.73·13-s − 0.200·17-s − 0.229·19-s + 1.25·23-s + 0.200·25-s + 1.20·29-s − 1.22·31-s − 0.140·35-s − 0.288·37-s − 0.571·41-s + 0.736·43-s + 0.704·47-s − 0.901·49-s − 1.24·53-s − 0.269·55-s − 1.77·59-s − 1.74·61-s + 0.774·65-s − 0.417·67-s − 0.613·71-s − 0.290·73-s + 0.188·77-s + 0.186·79-s + 1.46·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 0.828T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 6.48T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 + 1.75T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 4.82T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 + 13.6T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 3.41T + 67T^{2} \) |
| 71 | \( 1 + 5.17T + 71T^{2} \) |
| 73 | \( 1 + 2.48T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 6.48T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.145276517515521339880094838551, −7.43982365601928693943778971158, −6.91281080868657798020790165665, −6.02866999723893062138921468795, −4.84670366568486382460887858958, −4.64643592281404905008676085469, −3.43390971799688264911219369001, −2.59891451645837407559769890165, −1.45552072692072718068603756457, 0,
1.45552072692072718068603756457, 2.59891451645837407559769890165, 3.43390971799688264911219369001, 4.64643592281404905008676085469, 4.84670366568486382460887858958, 6.02866999723893062138921468795, 6.91281080868657798020790165665, 7.43982365601928693943778971158, 8.145276517515521339880094838551
