| L(s) = 1 | − 2·2-s + 4·4-s + 5-s − 7·7-s − 8·8-s − 2·10-s + 44·11-s − 66·13-s + 14·14-s + 16·16-s − 7·17-s − 4·19-s + 4·20-s − 88·22-s + 86·23-s − 124·25-s + 132·26-s − 28·28-s − 176·29-s + 162·31-s − 32·32-s + 14·34-s − 7·35-s − 199·37-s + 8·38-s − 8·40-s + 363·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.0894·5-s − 0.377·7-s − 0.353·8-s − 0.0632·10-s + 1.20·11-s − 1.40·13-s + 0.267·14-s + 1/4·16-s − 0.0998·17-s − 0.0482·19-s + 0.0447·20-s − 0.852·22-s + 0.779·23-s − 0.991·25-s + 0.995·26-s − 0.188·28-s − 1.12·29-s + 0.938·31-s − 0.176·32-s + 0.0706·34-s − 0.0338·35-s − 0.884·37-s + 0.0341·38-s − 0.0316·40-s + 1.38·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - T + p^{3} T^{2} \) |
| 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 66 T + p^{3} T^{2} \) |
| 17 | \( 1 + 7 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 T + p^{3} T^{2} \) |
| 23 | \( 1 - 86 T + p^{3} T^{2} \) |
| 29 | \( 1 + 176 T + p^{3} T^{2} \) |
| 31 | \( 1 - 162 T + p^{3} T^{2} \) |
| 37 | \( 1 + 199 T + p^{3} T^{2} \) |
| 41 | \( 1 - 363 T + p^{3} T^{2} \) |
| 43 | \( 1 + 451 T + p^{3} T^{2} \) |
| 47 | \( 1 - 9 T + p^{3} T^{2} \) |
| 53 | \( 1 + 174 T + p^{3} T^{2} \) |
| 59 | \( 1 + 587 T + p^{3} T^{2} \) |
| 61 | \( 1 + 156 T + p^{3} T^{2} \) |
| 67 | \( 1 + 560 T + p^{3} T^{2} \) |
| 71 | \( 1 + 532 T + p^{3} T^{2} \) |
| 73 | \( 1 + 854 T + p^{3} T^{2} \) |
| 79 | \( 1 + 747 T + p^{3} T^{2} \) |
| 83 | \( 1 + 613 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1266 T + p^{3} T^{2} \) |
| 97 | \( 1 - 64 T + p^{3} T^{2} \) |
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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
−10.23998494972030166532686649877, −9.533026112852872069233046025419, −8.852735676386272387503669708737, −7.60442949687864534788333956386, −6.85915885032037728999264576853, −5.83202112566545823659617074467, −4.43447861842606633627014415804, −3.03316104694181115699210780651, −1.64423259205725316536322945214, 0,
1.64423259205725316536322945214, 3.03316104694181115699210780651, 4.43447861842606633627014415804, 5.83202112566545823659617074467, 6.85915885032037728999264576853, 7.60442949687864534788333956386, 8.852735676386272387503669708737, 9.533026112852872069233046025419, 10.23998494972030166532686649877
