| L(s) = 1 | + 7.29·5-s − 5.87·7-s − 51.1·11-s − 13·13-s + 73.7·17-s − 59.9·19-s + 69.8·23-s − 71.8·25-s + 294.·29-s + 334.·31-s − 42.8·35-s + 261.·37-s − 222.·41-s − 79.2·43-s − 584.·47-s − 308.·49-s − 465.·53-s − 373.·55-s − 530.·59-s + 548.·61-s − 94.7·65-s + 384.·67-s − 307.·71-s − 844.·73-s + 300.·77-s − 30.1·79-s + 19.5·83-s + ⋯ |
| L(s) = 1 | + 0.652·5-s − 0.317·7-s − 1.40·11-s − 0.277·13-s + 1.05·17-s − 0.723·19-s + 0.633·23-s − 0.574·25-s + 1.88·29-s + 1.93·31-s − 0.206·35-s + 1.16·37-s − 0.848·41-s − 0.281·43-s − 1.81·47-s − 0.899·49-s − 1.20·53-s − 0.914·55-s − 1.16·59-s + 1.15·61-s − 0.180·65-s + 0.701·67-s − 0.513·71-s − 1.35·73-s + 0.444·77-s − 0.0429·79-s + 0.0258·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 5 | \( 1 - 7.29T + 125T^{2} \) |
| 7 | \( 1 + 5.87T + 343T^{2} \) |
| 11 | \( 1 + 51.1T + 1.33e3T^{2} \) |
| 17 | \( 1 - 73.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 59.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 69.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 294.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 334.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 261.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 222.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 79.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 584.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 465.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 530.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 548.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 384.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 307.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 844.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 30.1T + 4.93e5T^{2} \) |
| 83 | \( 1 - 19.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 513.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 787.T + 9.12e5T^{2} \) |
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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
−8.164890316442592924970392597251, −8.030058552080768642537252782994, −6.71003736748643046145230469162, −6.16397913383407530949936006017, −5.17585544328762991393188149535, −4.58520431917712232589822190787, −3.11533308893695196921795150548, −2.57113665196715262395150298148, −1.29144325965472644963121988852, 0,
1.29144325965472644963121988852, 2.57113665196715262395150298148, 3.11533308893695196921795150548, 4.58520431917712232589822190787, 5.17585544328762991393188149535, 6.16397913383407530949936006017, 6.71003736748643046145230469162, 8.030058552080768642537252782994, 8.164890316442592924970392597251
