| L(s) = 1 | + 5-s − 3.46·7-s + 0.732·11-s + 13-s − 0.535·17-s + 0.732·19-s − 2.73·23-s + 25-s − 2.53·29-s + 10.1·31-s − 3.46·35-s − 7.46·41-s − 10.7·43-s − 11.4·47-s + 4.99·49-s + 7.46·53-s + 0.732·55-s + 11.6·59-s + 8.39·61-s + 65-s + 0.928·67-s − 12.7·71-s + 10.9·73-s − 2.53·77-s − 13.4·79-s − 10.3·83-s − 0.535·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.30·7-s + 0.220·11-s + 0.277·13-s − 0.129·17-s + 0.167·19-s − 0.569·23-s + 0.200·25-s − 0.470·29-s + 1.83·31-s − 0.585·35-s − 1.16·41-s − 1.63·43-s − 1.67·47-s + 0.714·49-s + 1.02·53-s + 0.0987·55-s + 1.51·59-s + 1.07·61-s + 0.124·65-s + 0.113·67-s − 1.51·71-s + 1.27·73-s − 0.288·77-s − 1.51·79-s − 1.14·83-s − 0.0581·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 0.732T + 11T^{2} \) |
| 17 | \( 1 + 0.535T + 17T^{2} \) |
| 19 | \( 1 - 0.732T + 19T^{2} \) |
| 23 | \( 1 + 2.73T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 7.46T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 - 8.39T + 61T^{2} \) |
| 67 | \( 1 - 0.928T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−8.118960233764791075979175337318, −6.81714780658567347246555898434, −6.68732323160352695198740059906, −5.83838160373282176530536580978, −5.08870755327620197027634031775, −4.05137225822509980411689450593, −3.30888120019956100056072347290, −2.52384951584722811644545183493, −1.37595747114340262814393845487, 0,
1.37595747114340262814393845487, 2.52384951584722811644545183493, 3.30888120019956100056072347290, 4.05137225822509980411689450593, 5.08870755327620197027634031775, 5.83838160373282176530536580978, 6.68732323160352695198740059906, 6.81714780658567347246555898434, 8.118960233764791075979175337318
