L(s) = 1 | + 2-s + 4-s − 3.80·5-s + 0.515·7-s + 8-s − 3.80·10-s − 0.940·11-s − 2.83·13-s + 0.515·14-s + 16-s + 3.93·17-s − 3.57·19-s − 3.80·20-s − 0.940·22-s + 1.39·23-s + 9.49·25-s − 2.83·26-s + 0.515·28-s + 8.36·29-s + 2.17·31-s + 32-s + 3.93·34-s − 1.96·35-s + 3.67·37-s − 3.57·38-s − 3.80·40-s + 9.56·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.70·5-s + 0.194·7-s + 0.353·8-s − 1.20·10-s − 0.283·11-s − 0.786·13-s + 0.137·14-s + 0.250·16-s + 0.954·17-s − 0.819·19-s − 0.851·20-s − 0.200·22-s + 0.290·23-s + 1.89·25-s − 0.556·26-s + 0.0974·28-s + 1.55·29-s + 0.390·31-s + 0.176·32-s + 0.674·34-s − 0.331·35-s + 0.603·37-s − 0.579·38-s − 0.602·40-s + 1.49·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 - 0.515T + 7T^{2} \) |
| 11 | \( 1 + 0.940T + 11T^{2} \) |
| 13 | \( 1 + 2.83T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 + 3.57T + 19T^{2} \) |
| 23 | \( 1 - 1.39T + 23T^{2} \) |
| 29 | \( 1 - 8.36T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 - 3.67T + 37T^{2} \) |
| 41 | \( 1 - 9.56T + 41T^{2} \) |
| 43 | \( 1 + 9.00T + 43T^{2} \) |
| 47 | \( 1 + 2.89T + 47T^{2} \) |
| 53 | \( 1 + 1.17T + 53T^{2} \) |
| 59 | \( 1 + 12.3T + 59T^{2} \) |
| 61 | \( 1 - 2.12T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 2.35T + 71T^{2} \) |
| 73 | \( 1 - 10.1T + 73T^{2} \) |
| 79 | \( 1 + 5.14T + 79T^{2} \) |
| 83 | \( 1 - 3.57T + 83T^{2} \) |
| 89 | \( 1 - 0.00223T + 89T^{2} \) |
| 97 | \( 1 - 3.82T + 97T^{2} \) |
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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
−7.61674822588444203684077424863, −6.79345665425127049224314198843, −6.17232191859429650941651056552, −5.04859287621971765608638950550, −4.65782819692063898996004385691, −4.00433725008679726009817933946, −3.17022276823231056170974240188, −2.61441870260798722420396290280, −1.21275825187232792931119436913, 0,
1.21275825187232792931119436913, 2.61441870260798722420396290280, 3.17022276823231056170974240188, 4.00433725008679726009817933946, 4.65782819692063898996004385691, 5.04859287621971765608638950550, 6.17232191859429650941651056552, 6.79345665425127049224314198843, 7.61674822588444203684077424863