| L(s) = 1 | + 2.62e5·2-s − 1.15e9·3-s + 6.87e10·4-s − 3.59e11·5-s − 3.03e14·6-s − 6.58e15·7-s + 1.80e16·8-s + 8.92e17·9-s − 9.43e16·10-s + 6.37e18·11-s − 7.96e19·12-s + 5.56e20·13-s − 1.72e21·14-s + 4.16e20·15-s + 4.72e21·16-s + 9.53e22·17-s + 2.33e23·18-s − 3.47e23·19-s − 2.47e22·20-s + 7.62e24·21-s + 1.67e24·22-s + 1.10e25·23-s − 2.08e25·24-s − 7.26e25·25-s + 1.45e26·26-s − 5.12e26·27-s − 4.52e26·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.72·3-s + 0.5·4-s − 0.0421·5-s − 1.22·6-s − 1.52·7-s + 0.353·8-s + 1.98·9-s − 0.0298·10-s + 0.345·11-s − 0.863·12-s + 1.37·13-s − 1.08·14-s + 0.0728·15-s + 0.250·16-s + 1.64·17-s + 1.40·18-s − 0.765·19-s − 0.0210·20-s + 2.63·21-s + 0.244·22-s + 0.711·23-s − 0.610·24-s − 0.998·25-s + 0.970·26-s − 1.69·27-s − 0.764·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(38-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+37/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(19)\) |
\(\approx\) |
\(1.395186141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.395186141\) |
| \(L(\frac{39}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2.62e5T \) |
| good | 3 | \( 1 + 1.15e9T + 4.50e17T^{2} \) |
| 5 | \( 1 + 3.59e11T + 7.27e25T^{2} \) |
| 7 | \( 1 + 6.58e15T + 1.85e31T^{2} \) |
| 11 | \( 1 - 6.37e18T + 3.40e38T^{2} \) |
| 13 | \( 1 - 5.56e20T + 1.64e41T^{2} \) |
| 17 | \( 1 - 9.53e22T + 3.36e45T^{2} \) |
| 19 | \( 1 + 3.47e23T + 2.06e47T^{2} \) |
| 23 | \( 1 - 1.10e25T + 2.42e50T^{2} \) |
| 29 | \( 1 - 1.30e25T + 1.28e54T^{2} \) |
| 31 | \( 1 - 3.43e27T + 1.51e55T^{2} \) |
| 37 | \( 1 + 5.74e27T + 1.05e58T^{2} \) |
| 41 | \( 1 + 1.61e29T + 4.70e59T^{2} \) |
| 43 | \( 1 + 7.64e29T + 2.74e60T^{2} \) |
| 47 | \( 1 - 4.32e30T + 7.37e61T^{2} \) |
| 53 | \( 1 - 1.22e32T + 6.28e63T^{2} \) |
| 59 | \( 1 - 7.14e32T + 3.32e65T^{2} \) |
| 61 | \( 1 + 3.85e31T + 1.14e66T^{2} \) |
| 67 | \( 1 - 8.57e33T + 3.67e67T^{2} \) |
| 71 | \( 1 - 7.23e33T + 3.13e68T^{2} \) |
| 73 | \( 1 + 1.65e34T + 8.76e68T^{2} \) |
| 79 | \( 1 + 9.85e34T + 1.63e70T^{2} \) |
| 83 | \( 1 - 4.46e34T + 1.01e71T^{2} \) |
| 89 | \( 1 - 1.72e36T + 1.34e72T^{2} \) |
| 97 | \( 1 - 4.81e36T + 3.24e73T^{2} \) |
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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
−18.92295359697899073731511353687, −16.87358798325217309157824296804, −15.84735804575658006118092166087, −13.08662161694281413678592112086, −11.83403183625011824699565345154, −10.24672736015074998865242365012, −6.61171213408361419346297653220, −5.69411201142775513597962334611, −3.72453168370163521021637470905, −0.844811819902837591930642767390,
0.844811819902837591930642767390, 3.72453168370163521021637470905, 5.69411201142775513597962334611, 6.61171213408361419346297653220, 10.24672736015074998865242365012, 11.83403183625011824699565345154, 13.08662161694281413678592112086, 15.84735804575658006118092166087, 16.87358798325217309157824296804, 18.92295359697899073731511353687
