| L(s) = 1 | + 278.·2-s − 5.35e4·4-s + 3.90e5·5-s + 2.63e7·7-s − 5.14e7·8-s + 1.08e8·10-s + 1.05e9·11-s − 3.59e9·13-s + 7.33e9·14-s − 7.28e9·16-s + 1.03e10·17-s + 6.20e10·19-s − 2.09e10·20-s + 2.94e11·22-s − 2.58e11·23-s + 1.52e11·25-s − 9.99e11·26-s − 1.41e12·28-s − 2.00e12·29-s − 2.18e12·31-s + 4.70e12·32-s + 2.89e12·34-s + 1.02e13·35-s + 2.84e13·37-s + 1.72e13·38-s − 2.00e13·40-s − 6.05e13·41-s + ⋯ |
| L(s) = 1 | + 0.768·2-s − 0.408·4-s + 0.447·5-s + 1.72·7-s − 1.08·8-s + 0.343·10-s + 1.48·11-s − 1.22·13-s + 1.32·14-s − 0.424·16-s + 0.361·17-s + 0.838·19-s − 0.182·20-s + 1.14·22-s − 0.689·23-s + 0.200·25-s − 0.938·26-s − 0.706·28-s − 0.744·29-s − 0.459·31-s + 0.757·32-s + 0.277·34-s + 0.773·35-s + 1.33·37-s + 0.644·38-s − 0.484·40-s − 1.18·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(3.908907893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.908907893\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 3.90e5T \) |
| good | 2 | \( 1 - 278.T + 1.31e5T^{2} \) |
| 7 | \( 1 - 2.63e7T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.05e9T + 5.05e17T^{2} \) |
| 13 | \( 1 + 3.59e9T + 8.65e18T^{2} \) |
| 17 | \( 1 - 1.03e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 6.20e10T + 5.48e21T^{2} \) |
| 23 | \( 1 + 2.58e11T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.00e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 2.18e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.84e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 6.05e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 5.18e13T + 5.87e27T^{2} \) |
| 47 | \( 1 - 1.11e14T + 2.66e28T^{2} \) |
| 53 | \( 1 - 8.14e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.16e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 7.13e14T + 2.24e30T^{2} \) |
| 67 | \( 1 + 1.28e15T + 1.10e31T^{2} \) |
| 71 | \( 1 - 2.02e15T + 2.96e31T^{2} \) |
| 73 | \( 1 + 1.02e16T + 4.74e31T^{2} \) |
| 79 | \( 1 - 2.66e16T + 1.81e32T^{2} \) |
| 83 | \( 1 + 8.72e15T + 4.21e32T^{2} \) |
| 89 | \( 1 + 5.98e15T + 1.37e33T^{2} \) |
| 97 | \( 1 - 3.35e16T + 5.95e33T^{2} \) |
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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
−12.16612750446234157044136591153, −11.52940065269283355156489307683, −9.811638837128531969465237483089, −8.767722552198242623758680024166, −7.40917831509862513565999165427, −5.75428707972504658041568533992, −4.85005950530040044250162407252, −3.86915870111645501525374025457, −2.17261602300456250121523767879, −0.947060529802781346124883145759,
0.947060529802781346124883145759, 2.17261602300456250121523767879, 3.86915870111645501525374025457, 4.85005950530040044250162407252, 5.75428707972504658041568533992, 7.40917831509862513565999165427, 8.767722552198242623758680024166, 9.811638837128531969465237483089, 11.52940065269283355156489307683, 12.16612750446234157044136591153
