| L(s) = 1 | + 4.78·2-s + 14.8·4-s + 13.5·5-s − 19.0·7-s + 33.0·8-s + 64.8·10-s − 1.93·11-s + 38.4·13-s − 90.9·14-s + 38.7·16-s + 59.5·17-s + 83.6·19-s + 201.·20-s − 9.26·22-s + 170.·23-s + 58.7·25-s + 184.·26-s − 283.·28-s + 135.·29-s − 195.·31-s − 78.4·32-s + 285.·34-s − 257.·35-s − 52.4·37-s + 400.·38-s + 447.·40-s + 113.·41-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.86·4-s + 1.21·5-s − 1.02·7-s + 1.45·8-s + 2.05·10-s − 0.0530·11-s + 0.820·13-s − 1.73·14-s + 0.606·16-s + 0.850·17-s + 1.01·19-s + 2.25·20-s − 0.0897·22-s + 1.54·23-s + 0.470·25-s + 1.38·26-s − 1.91·28-s + 0.867·29-s − 1.13·31-s − 0.433·32-s + 1.43·34-s − 1.24·35-s − 0.232·37-s + 1.70·38-s + 1.76·40-s + 0.431·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(8.428179890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.428179890\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 241 | \( 1 - 241T \) |
| good | 2 | \( 1 - 4.78T + 8T^{2} \) |
| 5 | \( 1 - 13.5T + 125T^{2} \) |
| 7 | \( 1 + 19.0T + 343T^{2} \) |
| 11 | \( 1 + 1.93T + 1.33e3T^{2} \) |
| 13 | \( 1 - 38.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 83.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 170.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 195.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 52.4T + 5.06e4T^{2} \) |
| 41 | \( 1 - 113.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 2.63T + 7.95e4T^{2} \) |
| 47 | \( 1 + 418.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 283.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 739.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 259.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 408.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 269.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 512.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 928.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 448.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.36e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.77e3T + 9.12e5T^{2} \) |
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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
−8.898911090304191097386298360463, −7.56261682294508614879077016823, −6.63888616122477683295353006183, −6.23684773376699168759576837147, −5.41729158255948321674847964481, −4.97953633291881395882244028476, −3.56799534804382390104812845293, −3.22167839377340888396331410760, −2.22351494832738840911134338912, −1.06979948626740655779440650754,
1.06979948626740655779440650754, 2.22351494832738840911134338912, 3.22167839377340888396331410760, 3.56799534804382390104812845293, 4.97953633291881395882244028476, 5.41729158255948321674847964481, 6.23684773376699168759576837147, 6.63888616122477683295353006183, 7.56261682294508614879077016823, 8.898911090304191097386298360463
