| L(s) = 1 | − 3·3-s + 7.11·5-s + 8.88·7-s + 9·9-s − 26·11-s + 13·13-s − 21.3·15-s + 16.2·17-s − 35.5·19-s − 26.6·21-s + 153.·23-s − 74.3·25-s − 27·27-s − 223.·29-s + 126.·31-s + 78·33-s + 63.2·35-s − 217.·37-s − 39·39-s + 105.·41-s + 183.·43-s + 64.0·45-s + 96.6·47-s − 264.·49-s − 48.6·51-s − 386.·53-s − 184.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.636·5-s + 0.479·7-s + 0.333·9-s − 0.712·11-s + 0.277·13-s − 0.367·15-s + 0.231·17-s − 0.429·19-s − 0.276·21-s + 1.39·23-s − 0.595·25-s − 0.192·27-s − 1.43·29-s + 0.734·31-s + 0.411·33-s + 0.305·35-s − 0.966·37-s − 0.160·39-s + 0.403·41-s + 0.651·43-s + 0.212·45-s + 0.300·47-s − 0.769·49-s − 0.133·51-s − 1.00·53-s − 0.453·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 13 | \( 1 - 13T \) |
| good | 5 | \( 1 - 7.11T + 125T^{2} \) |
| 7 | \( 1 - 8.88T + 343T^{2} \) |
| 11 | \( 1 + 26T + 1.33e3T^{2} \) |
| 17 | \( 1 - 16.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 35.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 153.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 126.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 217.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 105.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 183.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 96.6T + 1.03e5T^{2} \) |
| 53 | \( 1 + 386.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 34.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 93.9T + 3.00e5T^{2} \) |
| 71 | \( 1 + 741.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 640.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 182.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 288.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 963.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 481.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.105393538583679537911402683128, −7.42906955836058059761279398717, −6.55110092600893619815788059161, −5.74820733241354185680904445960, −5.18667476648359443372132212885, −4.36969221592711796752941596261, −3.23061536413008528649680397516, −2.13841746579484245052465215851, −1.24811313124139847134225459235, 0,
1.24811313124139847134225459235, 2.13841746579484245052465215851, 3.23061536413008528649680397516, 4.36969221592711796752941596261, 5.18667476648359443372132212885, 5.74820733241354185680904445960, 6.55110092600893619815788059161, 7.42906955836058059761279398717, 8.105393538583679537911402683128
