| L(s) = 1 | + 10.5·5-s + 22·7-s − 5.29·11-s + 13·13-s − 116.·17-s + 126·19-s + 31.7·23-s − 12.9·25-s − 52.9·29-s + 182·31-s + 232.·35-s − 86·37-s + 444.·41-s − 96·43-s + 365.·47-s + 141·49-s + 190.·53-s − 56.0·55-s − 587.·59-s + 574·61-s + 137.·65-s + 530·67-s + 809.·71-s − 154·73-s − 116.·77-s + 460·79-s − 322.·83-s + ⋯ |
| L(s) = 1 | + 0.946·5-s + 1.18·7-s − 0.145·11-s + 0.277·13-s − 1.66·17-s + 1.52·19-s + 0.287·23-s − 0.103·25-s − 0.338·29-s + 1.05·31-s + 1.12·35-s − 0.382·37-s + 1.69·41-s − 0.340·43-s + 1.13·47-s + 0.411·49-s + 0.493·53-s − 0.137·55-s − 1.29·59-s + 1.20·61-s + 0.262·65-s + 0.966·67-s + 1.35·71-s − 0.246·73-s − 0.172·77-s + 0.655·79-s − 0.426·83-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
| \(L(2)\) |
\(\approx\) |
\(3.343467405\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.343467405\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 10.5T + 125T^{2} \) |
| 7 | \( 1 - 22T + 343T^{2} \) |
| 11 | \( 1 + 5.29T + 1.33e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 126T + 6.85e3T^{2} \) |
| 23 | \( 1 - 31.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 52.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 182T + 2.97e4T^{2} \) |
| 37 | \( 1 + 86T + 5.06e4T^{2} \) |
| 41 | \( 1 - 444.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 96T + 7.95e4T^{2} \) |
| 47 | \( 1 - 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 190.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 587.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 574T + 2.26e5T^{2} \) |
| 67 | \( 1 - 530T + 3.00e5T^{2} \) |
| 71 | \( 1 - 809.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 154T + 3.89e5T^{2} \) |
| 79 | \( 1 - 460T + 4.93e5T^{2} \) |
| 83 | \( 1 + 322.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.43e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 70T + 9.12e5T^{2} \) |
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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
−8.941501629264585522876707510875, −8.120352597034187055867383094680, −7.33773640224548675845314458046, −6.43721246041717929733553126462, −5.57205870389607757908118927437, −4.92186846850104143435437941243, −4.03407792790016390611768552350, −2.66442102580176422406613205546, −1.88532300025405019558503664857, −0.891026214816043831523261978795,
0.891026214816043831523261978795, 1.88532300025405019558503664857, 2.66442102580176422406613205546, 4.03407792790016390611768552350, 4.92186846850104143435437941243, 5.57205870389607757908118927437, 6.43721246041717929733553126462, 7.33773640224548675845314458046, 8.120352597034187055867383094680, 8.941501629264585522876707510875
