| L(s) = 1 | + 2.82·5-s + 7-s − 2·11-s − 0.828·13-s − 4·17-s + 19-s − 4.82·23-s + 3.00·25-s − 3.65·29-s + 6.82·31-s + 2.82·35-s − 3.17·37-s − 2·41-s − 9.65·43-s + 6.48·47-s + 49-s − 6·53-s − 5.65·55-s − 4·59-s − 6·61-s − 2.34·65-s − 11.3·67-s − 2.34·71-s + 10·73-s − 2·77-s + 9.17·79-s + 4.34·83-s + ⋯ |
| L(s) = 1 | + 1.26·5-s + 0.377·7-s − 0.603·11-s − 0.229·13-s − 0.970·17-s + 0.229·19-s − 1.00·23-s + 0.600·25-s − 0.679·29-s + 1.22·31-s + 0.478·35-s − 0.521·37-s − 0.312·41-s − 1.47·43-s + 0.945·47-s + 0.142·49-s − 0.824·53-s − 0.762·55-s − 0.520·59-s − 0.768·61-s − 0.290·65-s − 1.38·67-s − 0.278·71-s + 1.17·73-s − 0.227·77-s + 1.03·79-s + 0.476·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 0.828T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 23 | \( 1 + 4.82T + 23T^{2} \) |
| 29 | \( 1 + 3.65T + 29T^{2} \) |
| 31 | \( 1 - 6.82T + 31T^{2} \) |
| 37 | \( 1 + 3.17T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 9.65T + 43T^{2} \) |
| 47 | \( 1 - 6.48T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 2.34T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 - 4.34T + 83T^{2} \) |
| 89 | \( 1 - 0.343T + 89T^{2} \) |
| 97 | \( 1 + 0.343T + 97T^{2} \) |
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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
−7.37769003499262221919137583300, −6.45381371552092290498554303597, −6.09447244759399126547395746060, −5.22094896792251978155152435691, −4.80014796253553452130743610688, −3.85182797698426555506211960264, −2.80176277333637582187718452936, −2.13511727920923505958470566935, −1.46968303789891711414095216111, 0,
1.46968303789891711414095216111, 2.13511727920923505958470566935, 2.80176277333637582187718452936, 3.85182797698426555506211960264, 4.80014796253553452130743610688, 5.22094896792251978155152435691, 6.09447244759399126547395746060, 6.45381371552092290498554303597, 7.37769003499262221919137583300
