| L(s) = 1 | + 3.45·2-s + 3.95·4-s + 16.3·5-s + 31.8·7-s − 13.9·8-s + 56.4·10-s + 25.7·11-s + 56.5·13-s + 110.·14-s − 79.9·16-s − 20.6·17-s + 155.·19-s + 64.5·20-s + 89.0·22-s − 46.9·23-s + 141.·25-s + 195.·26-s + 125.·28-s − 78.4·29-s − 40.3·31-s − 164.·32-s − 71.4·34-s + 519.·35-s − 178.·37-s + 537.·38-s − 228.·40-s + 259.·41-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.494·4-s + 1.45·5-s + 1.71·7-s − 0.618·8-s + 1.78·10-s + 0.705·11-s + 1.20·13-s + 2.09·14-s − 1.24·16-s − 0.294·17-s + 1.87·19-s + 0.721·20-s + 0.862·22-s − 0.425·23-s + 1.13·25-s + 1.47·26-s + 0.848·28-s − 0.502·29-s − 0.233·31-s − 0.909·32-s − 0.360·34-s + 2.50·35-s − 0.791·37-s + 2.29·38-s − 0.902·40-s + 0.990·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.892397386\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.892397386\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 3.45T + 8T^{2} \) |
| 5 | \( 1 - 16.3T + 125T^{2} \) |
| 7 | \( 1 - 31.8T + 343T^{2} \) |
| 11 | \( 1 - 25.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 56.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 155.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 46.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 78.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 40.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 178.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 259.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 106.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 349.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 6.97T + 1.48e5T^{2} \) |
| 59 | \( 1 + 780.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 110.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 416.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 327.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 616.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 44.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 262.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.33e3T + 9.12e5T^{2} \) |
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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
−8.970971940660675821328279075007, −7.890479991808192796439798637152, −6.91248924150901155500399497572, −5.79149490376649437154281962847, −5.69413245645033081191959914911, −4.78014004730070271306147094240, −4.02079803326387128466146596848, −2.98329044444497077252981518854, −1.82861692632682523849700827714, −1.23416446959432170420492670462,
1.23416446959432170420492670462, 1.82861692632682523849700827714, 2.98329044444497077252981518854, 4.02079803326387128466146596848, 4.78014004730070271306147094240, 5.69413245645033081191959914911, 5.79149490376649437154281962847, 6.91248924150901155500399497572, 7.890479991808192796439798637152, 8.970971940660675821328279075007
