| L(s) = 1 | + (1 + i)2-s + 2i·4-s + (4.39 − 2.37i)5-s + (1.87 + 1.87i)7-s + (−2 + 2i)8-s + (6.77 + 2.01i)10-s + 14.7·11-s + (−13.2 + 13.2i)13-s + 3.74i·14-s − 4·16-s + (−1.17 − 1.17i)17-s + 15.1i·19-s + (4.75 + 8.79i)20-s + (14.7 + 14.7i)22-s + (22.5 − 22.5i)23-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.5i)2-s + 0.5i·4-s + (0.879 − 0.475i)5-s + (0.267 + 0.267i)7-s + (−0.250 + 0.250i)8-s + (0.677 + 0.201i)10-s + 1.34·11-s + (−1.02 + 1.02i)13-s + 0.267i·14-s − 0.250·16-s + (−0.0693 − 0.0693i)17-s + 0.796i·19-s + (0.237 + 0.439i)20-s + (0.671 + 0.671i)22-s + (0.978 − 0.978i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.498 - 0.867i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.498 - 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.965521465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.965521465\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1 - i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-4.39 + 2.37i)T \) |
| 7 | \( 1 + (-1.87 - 1.87i)T \) |
| good | 11 | \( 1 - 14.7T + 121T^{2} \) |
| 13 | \( 1 + (13.2 - 13.2i)T - 169iT^{2} \) |
| 17 | \( 1 + (1.17 + 1.17i)T + 289iT^{2} \) |
| 19 | \( 1 - 15.1iT - 361T^{2} \) |
| 23 | \( 1 + (-22.5 + 22.5i)T - 529iT^{2} \) |
| 29 | \( 1 - 3.29iT - 841T^{2} \) |
| 31 | \( 1 - 50.0T + 961T^{2} \) |
| 37 | \( 1 + (-6.15 - 6.15i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 35.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + (58.9 - 58.9i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-20.1 - 20.1i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (10.9 - 10.9i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 - 66.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 4.45T + 3.72e3T^{2} \) |
| 67 | \( 1 + (88.4 + 88.4i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 69.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-58.4 + 58.4i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 52.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-53.4 + 53.4i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 21.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-90.3 - 90.3i)T + 9.40e3iT^{2} \) |
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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
−10.44336658978278840600714384788, −9.381981287274140202131830033867, −8.948603936501544260509273861605, −7.84932010135346927342063524895, −6.62102218801977745644432750442, −6.19742159803936824170894879890, −4.89476633213580149783710753086, −4.35761082272571264070426069967, −2.73736489880998140443555531587, −1.44974996041468957682367598880,
1.07753796087596371092728020906, 2.40394514381248462536203518081, 3.40921525623649851663753774861, 4.69780498197917332129767735069, 5.55163479155864093057863697909, 6.59886612916411015306255723158, 7.32924750451540985528260165836, 8.751137759020853016343962837460, 9.656585101090757821753147524841, 10.22783152096793605453718584468
