| L(s) = 1 | + 0.381·2-s − 1.85·4-s − 5-s + 7-s − 1.47·8-s − 0.381·10-s + 4.23·11-s − 2.61·13-s + 0.381·14-s + 3.14·16-s − 0.854·17-s − 1.38·19-s + 1.85·20-s + 1.61·22-s + 2.14·23-s + 25-s − 26-s − 1.85·28-s + 5.85·29-s + 1.47·31-s + 4.14·32-s − 0.326·34-s − 35-s + 7.94·37-s − 0.527·38-s + 1.47·40-s + 12.5·41-s + ⋯ |
| L(s) = 1 | + 0.270·2-s − 0.927·4-s − 0.447·5-s + 0.377·7-s − 0.520·8-s − 0.120·10-s + 1.27·11-s − 0.726·13-s + 0.102·14-s + 0.786·16-s − 0.207·17-s − 0.317·19-s + 0.414·20-s + 0.344·22-s + 0.447·23-s + 0.200·25-s − 0.196·26-s − 0.350·28-s + 1.08·29-s + 0.264·31-s + 0.732·32-s − 0.0559·34-s − 0.169·35-s + 1.30·37-s − 0.0856·38-s + 0.232·40-s + 1.96·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.390553331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.390553331\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| good | 2 | \( 1 - 0.381T + 2T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 2.61T + 13T^{2} \) |
| 17 | \( 1 + 0.854T + 17T^{2} \) |
| 19 | \( 1 + 1.38T + 19T^{2} \) |
| 23 | \( 1 - 2.14T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 37 | \( 1 - 7.94T + 37T^{2} \) |
| 41 | \( 1 - 12.5T + 41T^{2} \) |
| 43 | \( 1 + 2.23T + 43T^{2} \) |
| 47 | \( 1 - 5.47T + 47T^{2} \) |
| 53 | \( 1 - 8.09T + 53T^{2} \) |
| 59 | \( 1 - 5.94T + 59T^{2} \) |
| 61 | \( 1 + 0.145T + 61T^{2} \) |
| 67 | \( 1 + 5.14T + 67T^{2} \) |
| 71 | \( 1 - 2.90T + 71T^{2} \) |
| 73 | \( 1 + 7.94T + 73T^{2} \) |
| 79 | \( 1 + 9.79T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 8.70T + 89T^{2} \) |
| 97 | \( 1 - 8.09T + 97T^{2} \) |
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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
−9.894894258491315620036767812393, −9.151349704947654767224788842367, −8.501199738456036070181491616443, −7.59125795509105494444134397025, −6.59916792494546717694452394608, −5.58867246235930635474461566436, −4.47603974804012323265464940027, −4.08658759880185269918886094901, −2.73827606529870622755387946045, −0.936636295880656901859652600956,
0.936636295880656901859652600956, 2.73827606529870622755387946045, 4.08658759880185269918886094901, 4.47603974804012323265464940027, 5.58867246235930635474461566436, 6.59916792494546717694452394608, 7.59125795509105494444134397025, 8.501199738456036070181491616443, 9.151349704947654767224788842367, 9.894894258491315620036767812393
