| L(s) = 1 | + i·3-s + (−2.20 − 0.353i)5-s + 1.65i·7-s − 9-s + 2.94·11-s + i·13-s + (0.353 − 2.20i)15-s + 1.46i·17-s − 1.65·21-s + 0.532i·23-s + (4.74 + 1.56i)25-s − i·27-s − 5.70·29-s + 2.94i·33-s + (0.585 − 3.65i)35-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.987 − 0.158i)5-s + 0.625i·7-s − 0.333·9-s + 0.888·11-s + 0.277i·13-s + (0.0913 − 0.570i)15-s + 0.355i·17-s − 0.361·21-s + 0.111i·23-s + (0.949 + 0.312i)25-s − 0.192i·27-s − 1.06·29-s + 0.513i·33-s + (0.0989 − 0.617i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.158i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7377326221\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7377326221\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (2.20 + 0.353i)T \) |
| 13 | \( 1 - iT \) |
| good | 7 | \( 1 - 1.65iT - 7T^{2} \) |
| 11 | \( 1 - 2.94T + 11T^{2} \) |
| 17 | \( 1 - 1.46iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 0.532iT - 23T^{2} \) |
| 29 | \( 1 + 5.70T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 8.77iT - 37T^{2} \) |
| 41 | \( 1 - 1.23T + 41T^{2} \) |
| 43 | \( 1 + 1.70iT - 43T^{2} \) |
| 47 | \( 1 + 2.70iT - 47T^{2} \) |
| 53 | \( 1 - 8.77iT - 53T^{2} \) |
| 59 | \( 1 + 3.83T + 59T^{2} \) |
| 61 | \( 1 + 0.241T + 61T^{2} \) |
| 67 | \( 1 + 2.58iT - 67T^{2} \) |
| 71 | \( 1 - 2.55T + 71T^{2} \) |
| 73 | \( 1 + 0.188iT - 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 7.91iT - 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 16.8iT - 97T^{2} \) |
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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
−9.001070303370423858731233496623, −8.445381598555037934251431496197, −7.64269891008128556659530537256, −6.80807708057565550024731463082, −5.97498212948710250945463195792, −5.11869646235009133898277361265, −4.25501030142273824511681645623, −3.69899002087318342918964948490, −2.74335704516143798182952387661, −1.39390351584221320972669031704,
0.24940741011716047740008863866, 1.34662723421781216754737869886, 2.64791759845230896278822645769, 3.69205776905851028041805369866, 4.19409316936841534857599166374, 5.29124959682847674155506313199, 6.23359435520670150153271271583, 7.09585046574799300320011285456, 7.41320652509797470731153080445, 8.221765798848655914291610943972
