| L(s) = 1 | + 5·7-s − 14·11-s + 13-s − 46·17-s + 19·19-s + 46·23-s − 14·29-s + 133·31-s + 258·37-s − 84·41-s − 167·43-s − 410·47-s − 318·49-s − 456·53-s + 194·59-s − 17·61-s + 653·67-s − 828·71-s + 570·73-s − 70·77-s − 552·79-s − 142·83-s + 1.10e3·89-s + 5·91-s + 841·97-s − 552·101-s − 308·103-s + ⋯ |
| L(s) = 1 | + 0.269·7-s − 0.383·11-s + 0.0213·13-s − 0.656·17-s + 0.229·19-s + 0.417·23-s − 0.0896·29-s + 0.770·31-s + 1.14·37-s − 0.319·41-s − 0.592·43-s − 1.27·47-s − 0.927·49-s − 1.18·53-s + 0.428·59-s − 0.0356·61-s + 1.19·67-s − 1.38·71-s + 0.913·73-s − 0.103·77-s − 0.786·79-s − 0.187·83-s + 1.31·89-s + 0.00575·91-s + 0.880·97-s − 0.543·101-s − 0.294·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5 T + p^{3} T^{2} \) |
| 11 | \( 1 + 14 T + p^{3} T^{2} \) |
| 13 | \( 1 - T + p^{3} T^{2} \) |
| 17 | \( 1 + 46 T + p^{3} T^{2} \) |
| 19 | \( 1 - p T + p^{3} T^{2} \) |
| 23 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 29 | \( 1 + 14 T + p^{3} T^{2} \) |
| 31 | \( 1 - 133 T + p^{3} T^{2} \) |
| 37 | \( 1 - 258 T + p^{3} T^{2} \) |
| 41 | \( 1 + 84 T + p^{3} T^{2} \) |
| 43 | \( 1 + 167 T + p^{3} T^{2} \) |
| 47 | \( 1 + 410 T + p^{3} T^{2} \) |
| 53 | \( 1 + 456 T + p^{3} T^{2} \) |
| 59 | \( 1 - 194 T + p^{3} T^{2} \) |
| 61 | \( 1 + 17 T + p^{3} T^{2} \) |
| 67 | \( 1 - 653 T + p^{3} T^{2} \) |
| 71 | \( 1 + 828 T + p^{3} T^{2} \) |
| 73 | \( 1 - 570 T + p^{3} T^{2} \) |
| 79 | \( 1 + 552 T + p^{3} T^{2} \) |
| 83 | \( 1 + 142 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1104 T + p^{3} T^{2} \) |
| 97 | \( 1 - 841 T + p^{3} T^{2} \) |
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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
−8.394062333003324819342546269676, −7.87622047089160165074485813137, −6.88344373330780905111651041619, −6.19406873968322098397404755407, −5.13901083687427881919391142961, −4.51277388064621196337264088417, −3.37899427814501525479819068000, −2.43147235528511051353944144449, −1.30291284351062520066448049432, 0,
1.30291284351062520066448049432, 2.43147235528511051353944144449, 3.37899427814501525479819068000, 4.51277388064621196337264088417, 5.13901083687427881919391142961, 6.19406873968322098397404755407, 6.88344373330780905111651041619, 7.87622047089160165074485813137, 8.394062333003324819342546269676
