L(s) = 1 | − 2-s + 2.73·3-s + 4-s − 1.10·5-s − 2.73·6-s + 2.49·7-s − 8-s + 4.47·9-s + 1.10·10-s + 3.92·11-s + 2.73·12-s − 0.165·13-s − 2.49·14-s − 3.02·15-s + 16-s + 5.98·17-s − 4.47·18-s + 4.00·19-s − 1.10·20-s + 6.82·21-s − 3.92·22-s + 4.73·23-s − 2.73·24-s − 3.77·25-s + 0.165·26-s + 4.02·27-s + 2.49·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.57·3-s + 0.5·4-s − 0.494·5-s − 1.11·6-s + 0.943·7-s − 0.353·8-s + 1.49·9-s + 0.349·10-s + 1.18·11-s + 0.789·12-s − 0.0459·13-s − 0.667·14-s − 0.779·15-s + 0.250·16-s + 1.45·17-s − 1.05·18-s + 0.918·19-s − 0.247·20-s + 1.48·21-s − 0.837·22-s + 0.987·23-s − 0.557·24-s − 0.755·25-s + 0.0324·26-s + 0.774·27-s + 0.471·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.052740197\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.052740197\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 2017 | \( 1 - T \) |
good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 + 1.10T + 5T^{2} \) |
| 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 - 3.92T + 11T^{2} \) |
| 13 | \( 1 + 0.165T + 13T^{2} \) |
| 17 | \( 1 - 5.98T + 17T^{2} \) |
| 19 | \( 1 - 4.00T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 - 6.81T + 29T^{2} \) |
| 31 | \( 1 - 2.17T + 31T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + 6.55T + 41T^{2} \) |
| 43 | \( 1 - 3.89T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 4.99T + 53T^{2} \) |
| 59 | \( 1 - 1.69T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 7.82T + 67T^{2} \) |
| 71 | \( 1 + 6.98T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 4.45T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.476488153421339864998726793918, −7.905586879549068840931148703853, −7.34101681134716802166863400698, −6.65855895518072107524593570875, −5.39683694715625963919144072033, −4.47182710260189558779677778026, −3.45200229144203768371505081967, −3.08400440006594234631454965430, −1.77612452000256839617857965456, −1.18089637888801265213455887469,
1.18089637888801265213455887469, 1.77612452000256839617857965456, 3.08400440006594234631454965430, 3.45200229144203768371505081967, 4.47182710260189558779677778026, 5.39683694715625963919144072033, 6.65855895518072107524593570875, 7.34101681134716802166863400698, 7.905586879549068840931148703853, 8.476488153421339864998726793918