| L(s) = 1 | + 2-s − 0.590·3-s + 4-s − 1.61·5-s − 0.590·6-s + 3.93·7-s + 8-s − 2.65·9-s − 1.61·10-s + 11-s − 0.590·12-s + 2.64·13-s + 3.93·14-s + 0.951·15-s + 16-s + 7.76·17-s − 2.65·18-s − 1.61·20-s − 2.32·21-s + 22-s + 5.05·23-s − 0.590·24-s − 2.40·25-s + 2.64·26-s + 3.33·27-s + 3.93·28-s + 2.32·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.340·3-s + 0.5·4-s − 0.720·5-s − 0.241·6-s + 1.48·7-s + 0.353·8-s − 0.883·9-s − 0.509·10-s + 0.301·11-s − 0.170·12-s + 0.734·13-s + 1.05·14-s + 0.245·15-s + 0.250·16-s + 1.88·17-s − 0.624·18-s − 0.360·20-s − 0.507·21-s + 0.213·22-s + 1.05·23-s − 0.120·24-s − 0.480·25-s + 0.519·26-s + 0.642·27-s + 0.743·28-s + 0.432·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.383540527\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.383540527\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.590T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 7 | \( 1 - 3.93T + 7T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 17 | \( 1 - 7.76T + 17T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 - 2.32T + 29T^{2} \) |
| 31 | \( 1 - 2.49T + 31T^{2} \) |
| 37 | \( 1 + 4.93T + 37T^{2} \) |
| 41 | \( 1 - 3.11T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 8.20T + 47T^{2} \) |
| 53 | \( 1 + 4.63T + 53T^{2} \) |
| 59 | \( 1 + 3.84T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 9.53T + 67T^{2} \) |
| 71 | \( 1 + 9.92T + 71T^{2} \) |
| 73 | \( 1 - 3.69T + 73T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 - 3.01T + 83T^{2} \) |
| 89 | \( 1 + 5.80T + 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−7.68213346101881853413996267804, −7.35140881910838009952478543350, −6.09730103476465897992494738518, −5.80344152078073792395188687674, −4.89873508404469546062909711810, −4.51392079213300009188318941498, −3.48414620706396544958625173185, −2.99216806870125743259748198876, −1.68938057149388482281423279847, −0.898229596006179263598435738851,
0.898229596006179263598435738851, 1.68938057149388482281423279847, 2.99216806870125743259748198876, 3.48414620706396544958625173185, 4.51392079213300009188318941498, 4.89873508404469546062909711810, 5.80344152078073792395188687674, 6.09730103476465897992494738518, 7.35140881910838009952478543350, 7.68213346101881853413996267804
