L(s) = 1 | − 2-s + 2.29·3-s + 4-s + 1.83·5-s − 2.29·6-s + 3.96·7-s − 8-s + 2.27·9-s − 1.83·10-s − 6.34·11-s + 2.29·12-s + 0.666·13-s − 3.96·14-s + 4.21·15-s + 16-s − 2.51·17-s − 2.27·18-s + 6.49·19-s + 1.83·20-s + 9.10·21-s + 6.34·22-s + 3.92·23-s − 2.29·24-s − 1.62·25-s − 0.666·26-s − 1.65·27-s + 3.96·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.32·3-s + 0.5·4-s + 0.821·5-s − 0.937·6-s + 1.49·7-s − 0.353·8-s + 0.759·9-s − 0.580·10-s − 1.91·11-s + 0.663·12-s + 0.184·13-s − 1.05·14-s + 1.08·15-s + 0.250·16-s − 0.610·17-s − 0.537·18-s + 1.49·19-s + 0.410·20-s + 1.98·21-s + 1.35·22-s + 0.818·23-s − 0.468·24-s − 0.325·25-s − 0.130·26-s − 0.318·27-s + 0.748·28-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
\(L(1)\) |
\(\approx\) |
\(2.990382709\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.990382709\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.29T + 3T^{2} \) |
| 5 | \( 1 - 1.83T + 5T^{2} \) |
| 7 | \( 1 - 3.96T + 7T^{2} \) |
| 11 | \( 1 + 6.34T + 11T^{2} \) |
| 13 | \( 1 - 0.666T + 13T^{2} \) |
| 17 | \( 1 + 2.51T + 17T^{2} \) |
| 19 | \( 1 - 6.49T + 19T^{2} \) |
| 23 | \( 1 - 3.92T + 23T^{2} \) |
| 29 | \( 1 - 6.27T + 29T^{2} \) |
| 31 | \( 1 - 2.49T + 31T^{2} \) |
| 37 | \( 1 + 1.13T + 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 0.0452T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 + 4.67T + 53T^{2} \) |
| 59 | \( 1 + 0.176T + 59T^{2} \) |
| 61 | \( 1 - 9.50T + 61T^{2} \) |
| 67 | \( 1 - 8.80T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 - 9.00T + 73T^{2} \) |
| 79 | \( 1 - 4.24T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 9.19T + 89T^{2} \) |
| 97 | \( 1 - 8.83T + 97T^{2} \) |
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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
−8.321336953033901111117776111016, −7.911853542211432659802496533717, −7.49079179989781008964206489345, −6.39156621602934091227922393341, −5.21136062601359029160505032652, −4.97141183731482002743152350027, −3.49927548556074361633677585372, −2.51427616214161910107809869397, −2.20158624419290736200874533559, −1.08633256473544823085833895259,
1.08633256473544823085833895259, 2.20158624419290736200874533559, 2.51427616214161910107809869397, 3.49927548556074361633677585372, 4.97141183731482002743152350027, 5.21136062601359029160505032652, 6.39156621602934091227922393341, 7.49079179989781008964206489345, 7.911853542211432659802496533717, 8.321336953033901111117776111016