| L(s) = 1 | − 3.25·5-s + 7·7-s + 57.6·11-s − 6.25·13-s + 75.3·17-s − 108.·19-s − 113.·23-s − 114.·25-s − 191.·29-s − 156.·31-s − 22.8·35-s + 215.·37-s + 133.·41-s − 122.·43-s − 194.·47-s + 49·49-s − 444.·53-s − 187.·55-s − 321.·59-s + 424.·61-s + 20.3·65-s + 1.02e3·67-s + 127.·71-s + 676.·73-s + 403.·77-s + 935.·79-s + 849.·83-s + ⋯ |
| L(s) = 1 | − 0.291·5-s + 0.377·7-s + 1.57·11-s − 0.133·13-s + 1.07·17-s − 1.30·19-s − 1.03·23-s − 0.915·25-s − 1.22·29-s − 0.907·31-s − 0.110·35-s + 0.955·37-s + 0.509·41-s − 0.433·43-s − 0.604·47-s + 0.142·49-s − 1.15·53-s − 0.460·55-s − 0.708·59-s + 0.890·61-s + 0.0389·65-s + 1.87·67-s + 0.213·71-s + 1.08·73-s + 0.597·77-s + 1.33·79-s + 1.12·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 3.25T + 125T^{2} \) |
| 11 | \( 1 - 57.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 6.25T + 2.19e3T^{2} \) |
| 17 | \( 1 - 75.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 108.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 156.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 215.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 133.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 194.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 444.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 321.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 424.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.02e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 127.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 676.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 935.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 849.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 21.8T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.84e3T + 9.12e5T^{2} \) |
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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
−8.207286901242899007303125514299, −7.82789376833158936624007947949, −6.75652749851291002137489372728, −6.10640103336212800994304733272, −5.20645218862724320887813837660, −3.99459989017392234106773971278, −3.75496754712172505001504388091, −2.20065169146678812396834189324, −1.34185922544227646441115197903, 0,
1.34185922544227646441115197903, 2.20065169146678812396834189324, 3.75496754712172505001504388091, 3.99459989017392234106773971278, 5.20645218862724320887813837660, 6.10640103336212800994304733272, 6.75652749851291002137489372728, 7.82789376833158936624007947949, 8.207286901242899007303125514299
