| L(s) = 1 | + 2.29·2-s − 27.4·3-s − 26.7·4-s + 25·5-s − 63.0·6-s − 30.5·7-s − 134.·8-s + 511.·9-s + 57.4·10-s + 121·11-s + 733.·12-s − 226.·13-s − 70.1·14-s − 686.·15-s + 545.·16-s + 52.5·17-s + 1.17e3·18-s + 361·19-s − 668.·20-s + 839.·21-s + 277.·22-s + 3.12e3·23-s + 3.70e3·24-s + 625·25-s − 520.·26-s − 7.36e3·27-s + 816.·28-s + ⋯ |
| L(s) = 1 | + 0.405·2-s − 1.76·3-s − 0.835·4-s + 0.447·5-s − 0.715·6-s − 0.235·7-s − 0.744·8-s + 2.10·9-s + 0.181·10-s + 0.301·11-s + 1.47·12-s − 0.372·13-s − 0.0956·14-s − 0.787·15-s + 0.532·16-s + 0.0441·17-s + 0.853·18-s + 0.229·19-s − 0.373·20-s + 0.415·21-s + 0.122·22-s + 1.23·23-s + 1.31·24-s + 0.200·25-s − 0.151·26-s − 1.94·27-s + 0.196·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8765680151\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8765680151\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 - 2.29T + 32T^{2} \) |
| 3 | \( 1 + 27.4T + 243T^{2} \) |
| 7 | \( 1 + 30.5T + 1.68e4T^{2} \) |
| 13 | \( 1 + 226.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 52.5T + 1.41e6T^{2} \) |
| 23 | \( 1 - 3.12e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.75e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.43e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.72e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 8.13e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.72e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.77e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.35e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.13e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.45e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.91e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.61e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.49e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.64e5T + 8.58e9T^{2} \) |
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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
−9.466609189646409620985132920272, −8.462541474008496116942339748717, −7.09859669682102302305036004759, −6.44047273969096019617725798807, −5.60063281979454286221461002611, −5.01797673222493468663439920122, −4.33526289904116647198912032802, −3.10949249338397537096035285317, −1.37877054878160703224880344011, −0.46320072153534287670135658558,
0.46320072153534287670135658558, 1.37877054878160703224880344011, 3.10949249338397537096035285317, 4.33526289904116647198912032802, 5.01797673222493468663439920122, 5.60063281979454286221461002611, 6.44047273969096019617725798807, 7.09859669682102302305036004759, 8.462541474008496116942339748717, 9.466609189646409620985132920272
