| L(s) = 1 | − 1.79·2-s + 1.20·4-s + 3.95·5-s − 4.84·7-s + 1.42·8-s − 7.08·10-s + 2.42·11-s − 0.0494·13-s + 8.67·14-s − 4.95·16-s + 6.29·17-s − 4.92·19-s + 4.77·20-s − 4.34·22-s − 23-s + 10.6·25-s + 0.0885·26-s − 5.84·28-s + 29-s − 0.565·31-s + 6.03·32-s − 11.2·34-s − 19.1·35-s − 0.214·37-s + 8.82·38-s + 5.61·40-s + 6.17·41-s + ⋯ |
| L(s) = 1 | − 1.26·2-s + 0.603·4-s + 1.76·5-s − 1.83·7-s + 0.502·8-s − 2.23·10-s + 0.732·11-s − 0.0137·13-s + 2.31·14-s − 1.23·16-s + 1.52·17-s − 1.13·19-s + 1.06·20-s − 0.927·22-s − 0.208·23-s + 2.12·25-s + 0.0173·26-s − 1.10·28-s + 0.185·29-s − 0.101·31-s + 1.06·32-s − 1.93·34-s − 3.23·35-s − 0.0353·37-s + 1.43·38-s + 0.888·40-s + 0.964·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.125461738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.125461738\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 5 | \( 1 - 3.95T + 5T^{2} \) |
| 7 | \( 1 + 4.84T + 7T^{2} \) |
| 11 | \( 1 - 2.42T + 11T^{2} \) |
| 13 | \( 1 + 0.0494T + 13T^{2} \) |
| 17 | \( 1 - 6.29T + 17T^{2} \) |
| 19 | \( 1 + 4.92T + 19T^{2} \) |
| 31 | \( 1 + 0.565T + 31T^{2} \) |
| 37 | \( 1 + 0.214T + 37T^{2} \) |
| 41 | \( 1 - 6.17T + 41T^{2} \) |
| 43 | \( 1 + 1.79T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 - 0.794T + 53T^{2} \) |
| 59 | \( 1 - 6.36T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 2.91T + 73T^{2} \) |
| 79 | \( 1 + 2.38T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 6.46T + 97T^{2} \) |
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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
−8.374808838061388584570483519525, −7.29217908497755149324505178062, −6.71162048565647719208697952647, −6.08189554181427575433211048848, −5.64750483623109919326609114020, −4.42545994082576809494684226350, −3.37407451155163743419289864590, −2.51281193632956347879139915965, −1.63934720131292488062644975001, −0.69783226091177181192598695650,
0.69783226091177181192598695650, 1.63934720131292488062644975001, 2.51281193632956347879139915965, 3.37407451155163743419289864590, 4.42545994082576809494684226350, 5.64750483623109919326609114020, 6.08189554181427575433211048848, 6.71162048565647719208697952647, 7.29217908497755149324505178062, 8.374808838061388584570483519525
