| L(s) = 1 | + 2.63·2-s + 2.23·3-s + 4.96·4-s − 5-s + 5.91·6-s − 7-s + 7.83·8-s + 2.01·9-s − 2.63·10-s + 2.72·11-s + 11.1·12-s + 1.32·13-s − 2.63·14-s − 2.23·15-s + 10.7·16-s + 5.16·17-s + 5.32·18-s + 4.00·19-s − 4.96·20-s − 2.23·21-s + 7.18·22-s + 3.31·23-s + 17.5·24-s + 25-s + 3.50·26-s − 2.20·27-s − 4.96·28-s + ⋯ |
| L(s) = 1 | + 1.86·2-s + 1.29·3-s + 2.48·4-s − 0.447·5-s + 2.41·6-s − 0.377·7-s + 2.77·8-s + 0.672·9-s − 0.834·10-s + 0.820·11-s + 3.21·12-s + 0.368·13-s − 0.705·14-s − 0.578·15-s + 2.68·16-s + 1.25·17-s + 1.25·18-s + 0.918·19-s − 1.11·20-s − 0.488·21-s + 1.53·22-s + 0.691·23-s + 3.58·24-s + 0.200·25-s + 0.687·26-s − 0.423·27-s − 0.939·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(11.38641765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.38641765\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 - T \) |
| good | 2 | \( 1 - 2.63T + 2T^{2} \) |
| 3 | \( 1 - 2.23T + 3T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 13 | \( 1 - 1.32T + 13T^{2} \) |
| 17 | \( 1 - 5.16T + 17T^{2} \) |
| 19 | \( 1 - 4.00T + 19T^{2} \) |
| 23 | \( 1 - 3.31T + 23T^{2} \) |
| 29 | \( 1 + 3.92T + 29T^{2} \) |
| 31 | \( 1 + 6.66T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 3.99T + 41T^{2} \) |
| 43 | \( 1 + 5.70T + 43T^{2} \) |
| 47 | \( 1 - 0.309T + 47T^{2} \) |
| 53 | \( 1 + 0.760T + 53T^{2} \) |
| 59 | \( 1 - 3.18T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 - 8.73T + 71T^{2} \) |
| 73 | \( 1 + 1.07T + 73T^{2} \) |
| 79 | \( 1 - 7.12T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 0.252T + 89T^{2} \) |
| 97 | \( 1 + 5.71T + 97T^{2} \) |
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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
−7.53177865208757716121141746872, −7.13988613612153710110996939866, −6.37699443456746900637562720110, −5.52056726222624972624907236391, −5.01576675388790444614134111827, −3.88989605065273826533669217394, −3.48367728518288635721941072330, −3.24423748916122452757930089489, −2.20572921366599130587573065115, −1.34611665321481991536726895697,
1.34611665321481991536726895697, 2.20572921366599130587573065115, 3.24423748916122452757930089489, 3.48367728518288635721941072330, 3.88989605065273826533669217394, 5.01576675388790444614134111827, 5.52056726222624972624907236391, 6.37699443456746900637562720110, 7.13988613612153710110996939866, 7.53177865208757716121141746872
