| L(s) = 1 | − 3.55·2-s − 0.389·3-s + 4.66·4-s − 1.72·5-s + 1.38·6-s − 21.1·7-s + 11.8·8-s − 26.8·9-s + 6.14·10-s + 23.7·11-s − 1.81·12-s − 30.1·13-s + 75.3·14-s + 0.672·15-s − 79.5·16-s + 51.6·17-s + 95.5·18-s + 29.2·19-s − 8.05·20-s + 8.24·21-s − 84.5·22-s − 93.4·23-s − 4.62·24-s − 122.·25-s + 107.·26-s + 20.9·27-s − 98.7·28-s + ⋯ |
| L(s) = 1 | − 1.25·2-s − 0.0748·3-s + 0.582·4-s − 0.154·5-s + 0.0942·6-s − 1.14·7-s + 0.524·8-s − 0.994·9-s + 0.194·10-s + 0.650·11-s − 0.0436·12-s − 0.643·13-s + 1.43·14-s + 0.0115·15-s − 1.24·16-s + 0.737·17-s + 1.25·18-s + 0.353·19-s − 0.0900·20-s + 0.0856·21-s − 0.818·22-s − 0.847·23-s − 0.0393·24-s − 0.976·25-s + 0.810·26-s + 0.149·27-s − 0.666·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2213417060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2213417060\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 + 3.55T + 8T^{2} \) |
| 3 | \( 1 + 0.389T + 27T^{2} \) |
| 5 | \( 1 + 1.72T + 125T^{2} \) |
| 7 | \( 1 + 21.1T + 343T^{2} \) |
| 11 | \( 1 - 23.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 30.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 51.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 29.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 93.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 59.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 113.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 68.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 53.9T + 6.89e4T^{2} \) |
| 47 | \( 1 + 455.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 662.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 457.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 606.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 428.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 139.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 481.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.10e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 624.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.12e3T + 9.12e5T^{2} \) |
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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
−9.124985028016210205224868040588, −8.034584550555362548598262814731, −7.69506765170852701767380528471, −6.56742151649755055837453146199, −6.00664888263662529412495569071, −4.84451085095217154429109685643, −3.69339809432795338971851987004, −2.79499945105508193691854945327, −1.55253354341545104181203428241, −0.27035579450452965897494216337,
0.27035579450452965897494216337, 1.55253354341545104181203428241, 2.79499945105508193691854945327, 3.69339809432795338971851987004, 4.84451085095217154429109685643, 6.00664888263662529412495569071, 6.56742151649755055837453146199, 7.69506765170852701767380528471, 8.034584550555362548598262814731, 9.124985028016210205224868040588
