| L(s) = 1 | − 2·2-s + 7·3-s + 4·4-s + 5·5-s − 14·6-s − 2·7-s − 8·8-s + 22·9-s − 10·10-s + 37·11-s + 28·12-s + 27·13-s + 4·14-s + 35·15-s + 16·16-s + 24·17-s − 44·18-s − 88·19-s + 20·20-s − 14·21-s − 74·22-s − 28·23-s − 56·24-s − 100·25-s − 54·26-s − 35·27-s − 8·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.34·3-s + 1/2·4-s + 0.447·5-s − 0.952·6-s − 0.107·7-s − 0.353·8-s + 0.814·9-s − 0.316·10-s + 1.01·11-s + 0.673·12-s + 0.576·13-s + 0.0763·14-s + 0.602·15-s + 1/4·16-s + 0.342·17-s − 0.576·18-s − 1.06·19-s + 0.223·20-s − 0.145·21-s − 0.717·22-s − 0.253·23-s − 0.476·24-s − 4/5·25-s − 0.407·26-s − 0.249·27-s − 0.0539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.624166883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.624166883\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p T \) |
| 29 | \( 1 + p T \) |
| good | 3 | \( 1 - 7 T + p^{3} T^{2} \) |
| 5 | \( 1 - p T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 37 T + p^{3} T^{2} \) |
| 13 | \( 1 - 27 T + p^{3} T^{2} \) |
| 17 | \( 1 - 24 T + p^{3} T^{2} \) |
| 19 | \( 1 + 88 T + p^{3} T^{2} \) |
| 23 | \( 1 + 28 T + p^{3} T^{2} \) |
| 31 | \( 1 + 143 T + p^{3} T^{2} \) |
| 37 | \( 1 + 360 T + p^{3} T^{2} \) |
| 41 | \( 1 - 386 T + p^{3} T^{2} \) |
| 43 | \( 1 - 381 T + p^{3} T^{2} \) |
| 47 | \( 1 + 103 T + p^{3} T^{2} \) |
| 53 | \( 1 + 431 T + p^{3} T^{2} \) |
| 59 | \( 1 - 288 T + p^{3} T^{2} \) |
| 61 | \( 1 + 840 T + p^{3} T^{2} \) |
| 67 | \( 1 + 180 T + p^{3} T^{2} \) |
| 71 | \( 1 - 706 T + p^{3} T^{2} \) |
| 73 | \( 1 - 716 T + p^{3} T^{2} \) |
| 79 | \( 1 - 931 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 642 T + p^{3} T^{2} \) |
| 97 | \( 1 - 486 T + p^{3} T^{2} \) |
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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
−14.58998537718026150922806420628, −13.84117907375106197617483725543, −12.52455964550183565640656650200, −10.95805904945508211319510224218, −9.565472086589684678057719823502, −8.881030747772264524685708726163, −7.75930740843732184131812329619, −6.24847972788402069715395522468, −3.66289541600251766455478983961, −1.92863672971255715890439607834,
1.92863672971255715890439607834, 3.66289541600251766455478983961, 6.24847972788402069715395522468, 7.75930740843732184131812329619, 8.881030747772264524685708726163, 9.565472086589684678057719823502, 10.95805904945508211319510224218, 12.52455964550183565640656650200, 13.84117907375106197617483725543, 14.58998537718026150922806420628
