| L(s) = 1 | − 3·3-s + 6.81·7-s + 9·9-s − 25.7·11-s − 32.1·13-s + 9.16·17-s + 99.2·19-s − 20.4·21-s − 61.2·23-s − 27·27-s + 17.1·29-s − 74.9·31-s + 77.3·33-s + 55.9·37-s + 96.4·39-s − 54.1·41-s − 33.9·43-s − 152.·47-s − 296.·49-s − 27.5·51-s − 26.7·53-s − 297.·57-s + 567.·59-s − 249.·61-s + 61.3·63-s + 987.·67-s + 183.·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.368·7-s + 0.333·9-s − 0.707·11-s − 0.686·13-s + 0.130·17-s + 1.19·19-s − 0.212·21-s − 0.555·23-s − 0.192·27-s + 0.110·29-s − 0.434·31-s + 0.408·33-s + 0.248·37-s + 0.396·39-s − 0.206·41-s − 0.120·43-s − 0.474·47-s − 0.864·49-s − 0.0755·51-s − 0.0692·53-s − 0.692·57-s + 1.25·59-s − 0.524·61-s + 0.122·63-s + 1.80·67-s + 0.320·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.388745476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.388745476\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 6.81T + 343T^{2} \) |
| 11 | \( 1 + 25.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 32.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 9.16T + 4.91e3T^{2} \) |
| 19 | \( 1 - 99.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 61.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 17.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 74.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 55.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 54.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 33.9T + 7.95e4T^{2} \) |
| 47 | \( 1 + 152.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 26.7T + 1.48e5T^{2} \) |
| 59 | \( 1 - 567.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 249.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 987.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 705.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 278.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.13e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 4.63T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.45e3T + 9.12e5T^{2} \) |
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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
−8.490566621055192646101573694116, −7.74823144211687158963430732333, −7.16589267493221764464259814326, −6.22421356853213424532486267033, −5.31283557780439143015794485055, −4.91336426893136838038695921241, −3.81300291479291666875457845918, −2.76540631124741428522505420303, −1.70842617984989824824280882176, −0.53877502345933591308760408869,
0.53877502345933591308760408869, 1.70842617984989824824280882176, 2.76540631124741428522505420303, 3.81300291479291666875457845918, 4.91336426893136838038695921241, 5.31283557780439143015794485055, 6.22421356853213424532486267033, 7.16589267493221764464259814326, 7.74823144211687158963430732333, 8.490566621055192646101573694116
