| L(s) = 1 | − 2-s + 4-s − 1.73·5-s − 3.46·7-s − 8-s + 1.73·10-s − 3.46·11-s − 5·13-s + 3.46·14-s + 16-s + 6.92·17-s + 3.46·19-s − 1.73·20-s + 3.46·22-s − 2.00·25-s + 5·26-s − 3.46·28-s + 3·29-s − 8·31-s − 32-s − 6.92·34-s + 5.99·35-s − 3.46·38-s + 1.73·40-s + 9·41-s + 6.92·43-s − 3.46·44-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.774·5-s − 1.30·7-s − 0.353·8-s + 0.547·10-s − 1.04·11-s − 1.38·13-s + 0.925·14-s + 0.250·16-s + 1.68·17-s + 0.794·19-s − 0.387·20-s + 0.738·22-s − 0.400·25-s + 0.980·26-s − 0.654·28-s + 0.557·29-s − 1.43·31-s − 0.176·32-s − 1.18·34-s + 1.01·35-s − 0.561·38-s + 0.273·40-s + 1.40·41-s + 1.05·43-s − 0.522·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 5T + 13T^{2} \) |
| 17 | \( 1 - 6.92T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 6.92T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 1.73T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 5.19T + 61T^{2} \) |
| 67 | \( 1 + 6.92T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 11T + 73T^{2} \) |
| 79 | \( 1 + 6.92T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 - 12.1T + 97T^{2} \) |
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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
−7.47900782391916158214554887520, −7.10353809540573226986192393982, −5.83875816678653473915124566885, −5.60445578443122761902147088598, −4.53673660915300750369001233961, −3.55305355602309646316770158524, −3.00190026917899984258030116397, −2.30544452058776229748213635745, −0.841842223174742119422435315781, 0,
0.841842223174742119422435315781, 2.30544452058776229748213635745, 3.00190026917899984258030116397, 3.55305355602309646316770158524, 4.53673660915300750369001233961, 5.60445578443122761902147088598, 5.83875816678653473915124566885, 7.10353809540573226986192393982, 7.47900782391916158214554887520
