| L(s) = 1 | + 3-s + 5-s + 2.24·7-s + 9-s − 0.941·11-s − 4.24·13-s + 15-s − 4·17-s − 6.61·19-s + 2.24·21-s − 2.49·23-s + 25-s + 27-s + 6.74·29-s − 31-s − 0.941·33-s + 2.24·35-s − 3.75·37-s − 4.24·39-s + 2.49·41-s + 1.88·43-s + 45-s + 1.05·47-s − 1.94·49-s − 4·51-s − 9.43·53-s − 0.941·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.850·7-s + 0.333·9-s − 0.283·11-s − 1.17·13-s + 0.258·15-s − 0.970·17-s − 1.51·19-s + 0.490·21-s − 0.520·23-s + 0.200·25-s + 0.192·27-s + 1.25·29-s − 0.179·31-s − 0.163·33-s + 0.380·35-s − 0.616·37-s − 0.680·39-s + 0.390·41-s + 0.287·43-s + 0.149·45-s + 0.154·47-s − 0.277·49-s − 0.560·51-s − 1.29·53-s − 0.126·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 2.24T + 7T^{2} \) |
| 11 | \( 1 + 0.941T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 6.61T + 19T^{2} \) |
| 23 | \( 1 + 2.49T + 23T^{2} \) |
| 29 | \( 1 - 6.74T + 29T^{2} \) |
| 37 | \( 1 + 3.75T + 37T^{2} \) |
| 41 | \( 1 - 2.49T + 41T^{2} \) |
| 43 | \( 1 - 1.88T + 43T^{2} \) |
| 47 | \( 1 - 1.05T + 47T^{2} \) |
| 53 | \( 1 + 9.43T + 53T^{2} \) |
| 59 | \( 1 - 1.42T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 - 6.74T + 67T^{2} \) |
| 71 | \( 1 + 5.42T + 71T^{2} \) |
| 73 | \( 1 - 1.19T + 73T^{2} \) |
| 79 | \( 1 + 3.43T + 79T^{2} \) |
| 83 | \( 1 - 2.94T + 83T^{2} \) |
| 89 | \( 1 + 9.07T + 89T^{2} \) |
| 97 | \( 1 - 9.11T + 97T^{2} \) |
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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
−7.73704645658155629178429835679, −6.82200023088330471181360508654, −6.31705039635982577649639241981, −5.28465206462395063982062785153, −4.62556040337524117604333903749, −4.14729927780943954937694103583, −2.86538879517345780971172548568, −2.27017485682445314917740441895, −1.57993665875642817096039327731, 0,
1.57993665875642817096039327731, 2.27017485682445314917740441895, 2.86538879517345780971172548568, 4.14729927780943954937694103583, 4.62556040337524117604333903749, 5.28465206462395063982062785153, 6.31705039635982577649639241981, 6.82200023088330471181360508654, 7.73704645658155629178429835679
