| L(s) = 1 | + 7.13e4·2-s + 3.08e7·3-s + 2.93e9·4-s − 8.25e10·5-s + 2.20e12·6-s + 1.14e13·7-s + 5.63e13·8-s + 3.34e14·9-s − 5.88e15·10-s − 2.40e15·11-s + 9.06e16·12-s − 2.01e17·13-s + 8.15e17·14-s − 2.54e18·15-s − 2.29e18·16-s + 1.09e19·17-s + 2.38e19·18-s − 1.42e19·19-s − 2.42e20·20-s + 3.52e20·21-s − 1.71e20·22-s − 3.85e18·23-s + 1.73e21·24-s + 2.15e21·25-s − 1.43e22·26-s − 8.74e21·27-s + 3.35e22·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s + 1.24·3-s + 1.36·4-s − 1.20·5-s + 1.91·6-s + 0.910·7-s + 0.565·8-s + 0.541·9-s − 1.86·10-s − 0.173·11-s + 1.69·12-s − 1.09·13-s + 1.40·14-s − 1.50·15-s − 0.496·16-s + 0.925·17-s + 0.832·18-s − 0.215·19-s − 1.65·20-s + 1.13·21-s − 0.266·22-s − 0.00301·23-s + 0.702·24-s + 0.461·25-s − 1.68·26-s − 0.569·27-s + 1.24·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(32-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+31/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(16)\) |
\(\approx\) |
\(4.090583564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.090583564\) |
| \(L(\frac{33}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 - 7.13e4T + 2.14e9T^{2} \) |
| 3 | \( 1 - 3.08e7T + 6.17e14T^{2} \) |
| 5 | \( 1 + 8.25e10T + 4.65e21T^{2} \) |
| 7 | \( 1 - 1.14e13T + 1.57e26T^{2} \) |
| 11 | \( 1 + 2.40e15T + 1.91e32T^{2} \) |
| 13 | \( 1 + 2.01e17T + 3.40e34T^{2} \) |
| 17 | \( 1 - 1.09e19T + 1.39e38T^{2} \) |
| 19 | \( 1 + 1.42e19T + 4.37e39T^{2} \) |
| 23 | \( 1 + 3.85e18T + 1.63e42T^{2} \) |
| 29 | \( 1 - 7.63e22T + 2.15e45T^{2} \) |
| 31 | \( 1 - 1.86e23T + 1.70e46T^{2} \) |
| 37 | \( 1 - 1.23e24T + 4.11e48T^{2} \) |
| 41 | \( 1 - 1.38e25T + 9.91e49T^{2} \) |
| 43 | \( 1 + 2.67e25T + 4.34e50T^{2} \) |
| 47 | \( 1 - 7.40e25T + 6.83e51T^{2} \) |
| 53 | \( 1 - 3.56e25T + 2.83e53T^{2} \) |
| 59 | \( 1 + 2.36e27T + 7.87e54T^{2} \) |
| 61 | \( 1 + 5.44e27T + 2.21e55T^{2} \) |
| 67 | \( 1 + 9.41e27T + 4.05e56T^{2} \) |
| 71 | \( 1 + 2.10e28T + 2.44e57T^{2} \) |
| 73 | \( 1 - 3.92e28T + 5.79e57T^{2} \) |
| 79 | \( 1 - 1.79e29T + 6.70e58T^{2} \) |
| 83 | \( 1 + 4.54e29T + 3.10e59T^{2} \) |
| 89 | \( 1 - 2.60e29T + 2.69e60T^{2} \) |
| 97 | \( 1 + 5.38e30T + 3.88e61T^{2} \) |
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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
−24.74601604359586371830940866206, −23.33111690817342398940431907316, −21.15899146928555972189445545945, −19.67496808881181391172802518323, −15.25153064420284295903901090958, −14.18215877875086330379784124974, −12.00980514724768876514313786268, −7.899175154495250600673870691710, −4.48743998605134772879277686601, −2.85476292460552992262642155636,
2.85476292460552992262642155636, 4.48743998605134772879277686601, 7.899175154495250600673870691710, 12.00980514724768876514313786268, 14.18215877875086330379784124974, 15.25153064420284295903901090958, 19.67496808881181391172802518323, 21.15899146928555972189445545945, 23.33111690817342398940431907316, 24.74601604359586371830940866206
