| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 5.85·5-s − 6·6-s + 28.5·7-s − 8·8-s + 9·9-s − 11.7·10-s − 2.56·11-s + 12·12-s + 82.9·13-s − 57.1·14-s + 17.5·15-s + 16·16-s − 74.8·17-s − 18·18-s + 34·19-s + 23.4·20-s + 85.6·21-s + 5.13·22-s + 15.1·23-s − 24·24-s − 90.6·25-s − 165.·26-s + 27·27-s + 114.·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.524·5-s − 0.408·6-s + 1.54·7-s − 0.353·8-s + 0.333·9-s − 0.370·10-s − 0.0703·11-s + 0.288·12-s + 1.77·13-s − 1.09·14-s + 0.302·15-s + 0.250·16-s − 1.06·17-s − 0.235·18-s + 0.410·19-s + 0.262·20-s + 0.890·21-s + 0.0497·22-s + 0.137·23-s − 0.204·24-s − 0.725·25-s − 1.25·26-s + 0.192·27-s + 0.771·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.322692999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.322692999\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 59 | \( 1 - 59T \) |
| good | 5 | \( 1 - 5.85T + 125T^{2} \) |
| 7 | \( 1 - 28.5T + 343T^{2} \) |
| 11 | \( 1 + 2.56T + 1.33e3T^{2} \) |
| 13 | \( 1 - 82.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 74.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 34T + 6.85e3T^{2} \) |
| 23 | \( 1 - 15.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 239.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 242.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 269.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 179.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 441.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 249.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 19.2T + 1.48e5T^{2} \) |
| 61 | \( 1 - 789.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 829.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 825.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 913.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.22e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 873.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 888.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 384.T + 9.12e5T^{2} \) |
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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
−11.00478231461511973422692794527, −10.11200340950897202232202702423, −8.824779134648160846813919739788, −8.537506757522985625002198102990, −7.52226873176453074277556292778, −6.37215331837829028975083721783, −5.16693076999975991205616861697, −3.80126073757737553701627632781, −2.17445645535342592666006834043, −1.26708412265135184728524806385,
1.26708412265135184728524806385, 2.17445645535342592666006834043, 3.80126073757737553701627632781, 5.16693076999975991205616861697, 6.37215331837829028975083721783, 7.52226873176453074277556292778, 8.537506757522985625002198102990, 8.824779134648160846813919739788, 10.11200340950897202232202702423, 11.00478231461511973422692794527
