| L(s) = 1 | + 0.108·2-s − 3-s − 1.98·4-s + 5-s − 0.108·6-s + 0.790·7-s − 0.433·8-s + 9-s + 0.108·10-s − 0.287·11-s + 1.98·12-s + 4.64·13-s + 0.0859·14-s − 15-s + 3.92·16-s + 5.99·17-s + 0.108·18-s + 1.56·19-s − 1.98·20-s − 0.790·21-s − 0.0312·22-s − 3.81·23-s + 0.433·24-s + 25-s + 0.504·26-s − 27-s − 1.57·28-s + ⋯ |
| L(s) = 1 | + 0.0768·2-s − 0.577·3-s − 0.994·4-s + 0.447·5-s − 0.0443·6-s + 0.298·7-s − 0.153·8-s + 0.333·9-s + 0.0343·10-s − 0.0866·11-s + 0.573·12-s + 1.28·13-s + 0.0229·14-s − 0.258·15-s + 0.982·16-s + 1.45·17-s + 0.0256·18-s + 0.359·19-s − 0.444·20-s − 0.172·21-s − 0.00665·22-s − 0.796·23-s + 0.0884·24-s + 0.200·25-s + 0.0989·26-s − 0.192·27-s − 0.297·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.718108206\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718108206\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 0.108T + 2T^{2} \) |
| 7 | \( 1 - 0.790T + 7T^{2} \) |
| 11 | \( 1 + 0.287T + 11T^{2} \) |
| 13 | \( 1 - 4.64T + 13T^{2} \) |
| 17 | \( 1 - 5.99T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 3.81T + 23T^{2} \) |
| 29 | \( 1 - 8.60T + 29T^{2} \) |
| 31 | \( 1 + 7.99T + 31T^{2} \) |
| 37 | \( 1 - 5.61T + 37T^{2} \) |
| 41 | \( 1 - 1.25T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 - 7.46T + 53T^{2} \) |
| 59 | \( 1 + 9.20T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 4.15T + 73T^{2} \) |
| 79 | \( 1 - 6.62T + 79T^{2} \) |
| 83 | \( 1 - 9.73T + 83T^{2} \) |
| 89 | \( 1 - 3.23T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−7.987651018277311785117790531055, −7.58508109857714419306250715645, −6.36853054355895296137249559002, −5.81079719331394020113310213429, −5.35061935614958802583678966132, −4.47043622110545582118805618901, −3.81330197723292101994915641931, −2.94662296961113755534185828895, −1.51081388531914954321604701508, −0.78643138518670558920105850785,
0.78643138518670558920105850785, 1.51081388531914954321604701508, 2.94662296961113755534185828895, 3.81330197723292101994915641931, 4.47043622110545582118805618901, 5.35061935614958802583678966132, 5.81079719331394020113310213429, 6.36853054355895296137249559002, 7.58508109857714419306250715645, 7.987651018277311785117790531055
