| L(s) = 1 | − 6.46·3-s − 3.00·5-s − 7·7-s + 14.7·9-s − 11·11-s − 29.1·13-s + 19.4·15-s + 31.7·17-s − 81.9·19-s + 45.2·21-s − 163.·23-s − 115.·25-s + 79.0·27-s − 93.6·29-s − 55.7·31-s + 71.0·33-s + 21.0·35-s + 344.·37-s + 188.·39-s + 36.0·41-s − 175.·43-s − 44.3·45-s − 364.·47-s + 49·49-s − 205.·51-s + 71.2·53-s + 33.0·55-s + ⋯ |
| L(s) = 1 | − 1.24·3-s − 0.268·5-s − 0.377·7-s + 0.546·9-s − 0.301·11-s − 0.621·13-s + 0.334·15-s + 0.453·17-s − 0.988·19-s + 0.470·21-s − 1.48·23-s − 0.927·25-s + 0.563·27-s − 0.599·29-s − 0.322·31-s + 0.374·33-s + 0.101·35-s + 1.53·37-s + 0.773·39-s + 0.137·41-s − 0.622·43-s − 0.146·45-s − 1.13·47-s + 0.142·49-s − 0.564·51-s + 0.184·53-s + 0.0810·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5030388364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5030388364\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 + 6.46T + 27T^{2} \) |
| 5 | \( 1 + 3.00T + 125T^{2} \) |
| 13 | \( 1 + 29.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 31.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 81.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 163.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 93.6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 55.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 344.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 36.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 175.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 364.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 71.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 454.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 347.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 381.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 605.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 664.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 70.8T + 4.93e5T^{2} \) |
| 83 | \( 1 - 375.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 808.T + 9.12e5T^{2} \) |
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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
−10.24456048895978241834145162686, −9.686061423008184340908037654633, −8.340971355084247973682191981906, −7.49218565482856623943246694395, −6.39190131572731178795871625003, −5.77968830155796660041921987094, −4.78868412263933812378928304156, −3.73524586636528176218300335826, −2.18373343349134501432629898457, −0.42574170301776120103939541110,
0.42574170301776120103939541110, 2.18373343349134501432629898457, 3.73524586636528176218300335826, 4.78868412263933812378928304156, 5.77968830155796660041921987094, 6.39190131572731178795871625003, 7.49218565482856623943246694395, 8.340971355084247973682191981906, 9.686061423008184340908037654633, 10.24456048895978241834145162686
