| L(s) = 1 | + 7.21·3-s + 5·5-s − 7.21·7-s + 24.9·9-s + 43.2·11-s − 34·13-s + 36.0·15-s + 114·17-s − 51.9·21-s + 209.·23-s + 25·25-s − 14.4·27-s + 26·29-s − 100.·31-s + 312·33-s − 36.0·35-s + 150·37-s − 245.·39-s + 342·41-s − 454.·43-s + 124.·45-s − 584.·47-s − 291·49-s + 822.·51-s + 262·53-s + 216.·55-s − 490.·59-s + ⋯ |
| L(s) = 1 | + 1.38·3-s + 0.447·5-s − 0.389·7-s + 0.925·9-s + 1.18·11-s − 0.725·13-s + 0.620·15-s + 1.62·17-s − 0.540·21-s + 1.89·23-s + 0.200·25-s − 0.102·27-s + 0.166·29-s − 0.584·31-s + 1.64·33-s − 0.174·35-s + 0.666·37-s − 1.00·39-s + 1.30·41-s − 1.61·43-s + 0.414·45-s − 1.81·47-s − 0.848·49-s + 2.25·51-s + 0.679·53-s + 0.530·55-s − 1.08·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.336807114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.336807114\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 3 | \( 1 - 7.21T + 27T^{2} \) |
| 7 | \( 1 + 7.21T + 343T^{2} \) |
| 11 | \( 1 - 43.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34T + 2.19e3T^{2} \) |
| 17 | \( 1 - 114T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 - 209.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 26T + 2.43e4T^{2} \) |
| 31 | \( 1 + 100.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 150T + 5.06e4T^{2} \) |
| 41 | \( 1 - 342T + 6.89e4T^{2} \) |
| 43 | \( 1 + 454.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 584.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 262T + 1.48e5T^{2} \) |
| 59 | \( 1 + 490.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 262T + 2.26e5T^{2} \) |
| 67 | \( 1 + 497.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 682T + 3.89e5T^{2} \) |
| 79 | \( 1 - 201.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 151.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 630T + 7.04e5T^{2} \) |
| 97 | \( 1 + 966T + 9.12e5T^{2} \) |
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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
−11.17232895040403423161633032483, −9.694558696368641783800383550835, −9.521872490500277563366716824698, −8.470893821883778292385435232823, −7.48682571610062326990080666954, −6.51600209933778470726675759750, −5.11170736274355173469758393035, −3.62460652763853189192581789228, −2.81491063073298123861461612906, −1.36986175199117627351380614427,
1.36986175199117627351380614427, 2.81491063073298123861461612906, 3.62460652763853189192581789228, 5.11170736274355173469758393035, 6.51600209933778470726675759750, 7.48682571610062326990080666954, 8.470893821883778292385435232823, 9.521872490500277563366716824698, 9.694558696368641783800383550835, 11.17232895040403423161633032483
