| L(s) = 1 | + 8.86·3-s − 18.8·5-s + 20.8·7-s + 51.5·9-s + 13.7·11-s − 13·13-s − 167.·15-s + 6.86·17-s − 52.6·19-s + 184.·21-s + 188.·23-s + 230.·25-s + 218.·27-s + 206.·29-s − 290.·31-s + 121.·33-s − 393.·35-s + 220.·37-s − 115.·39-s + 149.·41-s + 183.·43-s − 973.·45-s + 146.·47-s + 92.3·49-s + 60.8·51-s − 409.·53-s − 259.·55-s + ⋯ |
| L(s) = 1 | + 1.70·3-s − 1.68·5-s + 1.12·7-s + 1.91·9-s + 0.376·11-s − 0.277·13-s − 2.87·15-s + 0.0979·17-s − 0.635·19-s + 1.92·21-s + 1.71·23-s + 1.84·25-s + 1.55·27-s + 1.32·29-s − 1.68·31-s + 0.642·33-s − 1.90·35-s + 0.978·37-s − 0.473·39-s + 0.570·41-s + 0.649·43-s − 3.22·45-s + 0.454·47-s + 0.269·49-s + 0.167·51-s − 1.06·53-s − 0.635·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.476029441\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.476029441\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 - 8.86T + 27T^{2} \) |
| 5 | \( 1 + 18.8T + 125T^{2} \) |
| 7 | \( 1 - 20.8T + 343T^{2} \) |
| 11 | \( 1 - 13.7T + 1.33e3T^{2} \) |
| 17 | \( 1 - 6.86T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 188.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 206.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 290.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 220.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 149.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 183.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 146.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 31.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 674.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 451.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 247.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 472.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 478.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 519.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 411.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−9.468671721650295770262551848253, −8.717674755498310342208874212251, −8.160277009108984750437551676729, −7.57514994653151233295522249873, −6.89502922470045501624798292242, −4.90912501834665528448193859999, −4.17755483067622585459035232593, −3.40810072306913503094537212447, −2.37874911817316611641753357554, −1.00962720255488920030645464845,
1.00962720255488920030645464845, 2.37874911817316611641753357554, 3.40810072306913503094537212447, 4.17755483067622585459035232593, 4.90912501834665528448193859999, 6.89502922470045501624798292242, 7.57514994653151233295522249873, 8.160277009108984750437551676729, 8.717674755498310342208874212251, 9.468671721650295770262551848253
