L(s) = 1 | + 3·3-s − 8·4-s − 5·5-s + 7·7-s + 9·9-s + 12·11-s − 24·12-s − 42·13-s − 15·15-s + 64·16-s + 26·17-s − 26·19-s + 40·20-s + 21·21-s + 23·23-s + 25·25-s + 27·27-s − 56·28-s − 82·29-s − 64·31-s + 36·33-s − 35·35-s − 72·36-s − 256·37-s − 126·39-s + 126·41-s + 508·43-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.328·11-s − 0.577·12-s − 0.896·13-s − 0.258·15-s + 16-s + 0.370·17-s − 0.313·19-s + 0.447·20-s + 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.525·29-s − 0.370·31-s + 0.189·33-s − 0.169·35-s − 1/3·36-s − 1.13·37-s − 0.517·39-s + 0.479·41-s + 1.80·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2415 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| 23 | \( 1 - p T \) |
good | 2 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 42 T + p^{3} T^{2} \) |
| 17 | \( 1 - 26 T + p^{3} T^{2} \) |
| 19 | \( 1 + 26 T + p^{3} T^{2} \) |
| 29 | \( 1 + 82 T + p^{3} T^{2} \) |
| 31 | \( 1 + 64 T + p^{3} T^{2} \) |
| 37 | \( 1 + 256 T + p^{3} T^{2} \) |
| 41 | \( 1 - 126 T + p^{3} T^{2} \) |
| 43 | \( 1 - 508 T + p^{3} T^{2} \) |
| 47 | \( 1 - 4 T + p^{3} T^{2} \) |
| 53 | \( 1 + 102 T + p^{3} T^{2} \) |
| 59 | \( 1 + 478 T + p^{3} T^{2} \) |
| 61 | \( 1 - 462 T + p^{3} T^{2} \) |
| 67 | \( 1 - 724 T + p^{3} T^{2} \) |
| 71 | \( 1 + 90 T + p^{3} T^{2} \) |
| 73 | \( 1 + 82 T + p^{3} T^{2} \) |
| 79 | \( 1 - 798 T + p^{3} T^{2} \) |
| 83 | \( 1 + 748 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1122 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1452 T + p^{3} T^{2} \) |
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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
−8.200289114431233876297739417393, −7.67616048263653292634182422222, −6.90834781986398171924302353013, −5.67344209073177794106420747519, −4.90984358455714755346091494091, −4.14713625880723693740180703985, −3.47508077383348727055863122344, −2.35663240155113101346399001756, −1.13998601751280699736553606232, 0,
1.13998601751280699736553606232, 2.35663240155113101346399001756, 3.47508077383348727055863122344, 4.14713625880723693740180703985, 4.90984358455714755346091494091, 5.67344209073177794106420747519, 6.90834781986398171924302353013, 7.67616048263653292634182422222, 8.200289114431233876297739417393