| L(s) = 1 | + 1.29·2-s − 0.320·4-s + (−1.01 + 1.99i)5-s + (2.26 − 1.37i)7-s − 3.00·8-s + (−1.31 + 2.58i)10-s + 3.12i·11-s + 2.45·13-s + (2.93 − 1.77i)14-s − 3.25·16-s + 4.43i·17-s + 4.17i·19-s + (0.324 − 0.639i)20-s + 4.04i·22-s − 5.77·23-s + ⋯ |
| L(s) = 1 | + 0.916·2-s − 0.160·4-s + (−0.452 + 0.891i)5-s + (0.855 − 0.518i)7-s − 1.06·8-s + (−0.414 + 0.817i)10-s + 0.941i·11-s + 0.681·13-s + (0.783 − 0.474i)14-s − 0.814·16-s + 1.07i·17-s + 0.957i·19-s + (0.0725 − 0.142i)20-s + 0.862i·22-s − 1.20·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0750 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 945 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0750 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22071 + 1.31602i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22071 + 1.31602i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.01 - 1.99i)T \) |
| 7 | \( 1 + (-2.26 + 1.37i)T \) |
| good | 2 | \( 1 - 1.29T + 2T^{2} \) |
| 11 | \( 1 - 3.12iT - 11T^{2} \) |
| 13 | \( 1 - 2.45T + 13T^{2} \) |
| 17 | \( 1 - 4.43iT - 17T^{2} \) |
| 19 | \( 1 - 4.17iT - 19T^{2} \) |
| 23 | \( 1 + 5.77T + 23T^{2} \) |
| 29 | \( 1 - 0.339iT - 29T^{2} \) |
| 31 | \( 1 - 4.40iT - 31T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 - 5.43T + 41T^{2} \) |
| 43 | \( 1 + 0.439iT - 43T^{2} \) |
| 47 | \( 1 - 10.2iT - 47T^{2} \) |
| 53 | \( 1 + 2.76T + 53T^{2} \) |
| 59 | \( 1 - 5.00T + 59T^{2} \) |
| 61 | \( 1 + 9.34iT - 61T^{2} \) |
| 67 | \( 1 + 15.7iT - 67T^{2} \) |
| 71 | \( 1 + 10.1iT - 71T^{2} \) |
| 73 | \( 1 + 9.26T + 73T^{2} \) |
| 79 | \( 1 - 15.6T + 79T^{2} \) |
| 83 | \( 1 - 4.13iT - 83T^{2} \) |
| 89 | \( 1 + 5.20T + 89T^{2} \) |
| 97 | \( 1 + 6.09T + 97T^{2} \) |
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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
−10.43450078269363057409808685904, −9.607422191424512079652023260041, −8.257748073704323279292629432890, −7.85956341439310192928051857804, −6.62378566491659729436666698971, −5.96978196248781168088993940340, −4.73338387906308853908396697566, −4.06623288118598191784678393547, −3.29891440545933672831322840024, −1.78477473621389011045963686719,
0.64411396971675645677586426936, 2.46985056531771263637016935314, 3.79560032404917154511427926933, 4.46656542419667500941843862808, 5.45668641375457980352020726200, 5.84345151479433257989776632449, 7.32243261023599709112106038561, 8.406664410561952038324920037634, 8.791653898477847762470283138648, 9.593707243482430741803180225757
