| L(s) = 1 | − 2-s + 1.94·3-s + 4-s + 3.30·5-s − 1.94·6-s + 7-s − 8-s + 0.788·9-s − 3.30·10-s + 5.79·11-s + 1.94·12-s + 4.91·13-s − 14-s + 6.43·15-s + 16-s − 7.24·17-s − 0.788·18-s + 4.79·19-s + 3.30·20-s + 1.94·21-s − 5.79·22-s − 3.99·23-s − 1.94·24-s + 5.91·25-s − 4.91·26-s − 4.30·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.12·3-s + 0.5·4-s + 1.47·5-s − 0.794·6-s + 0.377·7-s − 0.353·8-s + 0.262·9-s − 1.04·10-s + 1.74·11-s + 0.561·12-s + 1.36·13-s − 0.267·14-s + 1.66·15-s + 0.250·16-s − 1.75·17-s − 0.185·18-s + 1.10·19-s + 0.738·20-s + 0.424·21-s − 1.23·22-s − 0.832·23-s − 0.397·24-s + 1.18·25-s − 0.964·26-s − 0.828·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.586540319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.586540319\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 1.94T + 3T^{2} \) |
| 5 | \( 1 - 3.30T + 5T^{2} \) |
| 11 | \( 1 - 5.79T + 11T^{2} \) |
| 13 | \( 1 - 4.91T + 13T^{2} \) |
| 17 | \( 1 + 7.24T + 17T^{2} \) |
| 19 | \( 1 - 4.79T + 19T^{2} \) |
| 23 | \( 1 + 3.99T + 23T^{2} \) |
| 29 | \( 1 + 1.92T + 29T^{2} \) |
| 31 | \( 1 + 1.90T + 31T^{2} \) |
| 37 | \( 1 + 4.37T + 37T^{2} \) |
| 41 | \( 1 - 8.68T + 41T^{2} \) |
| 43 | \( 1 + 7.23T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 + 1.16T + 53T^{2} \) |
| 59 | \( 1 - 8.64T + 59T^{2} \) |
| 61 | \( 1 - 7.35T + 61T^{2} \) |
| 67 | \( 1 - 3.06T + 67T^{2} \) |
| 71 | \( 1 - 0.887T + 71T^{2} \) |
| 73 | \( 1 - 2.87T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 + 1.96T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−8.564899256542328977236712077711, −7.43587420652903136424985988488, −6.69197653843284779653363821626, −6.12074296932399421090907959796, −5.47724841434984157729090834290, −4.13946446226263921714096183283, −3.55207531519386334248155190007, −2.42559355751149795237907208382, −1.86300072896936156376187952988, −1.16363966326886676885246200447,
1.16363966326886676885246200447, 1.86300072896936156376187952988, 2.42559355751149795237907208382, 3.55207531519386334248155190007, 4.13946446226263921714096183283, 5.47724841434984157729090834290, 6.12074296932399421090907959796, 6.69197653843284779653363821626, 7.43587420652903136424985988488, 8.564899256542328977236712077711
