| L(s) = 1 | + 2-s − 4-s + (1 − 2i)5-s − 3·8-s + (1 − 2i)10-s − 2i·11-s + (3 − 2i)13-s − 16-s − 6i·19-s + (−1 + 2i)20-s − 2i·22-s − 6i·23-s + (−3 − 4i)25-s + (3 − 2i)26-s − 6·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.5·4-s + (0.447 − 0.894i)5-s − 1.06·8-s + (0.316 − 0.632i)10-s − 0.603i·11-s + (0.832 − 0.554i)13-s − 0.250·16-s − 1.37i·19-s + (−0.223 + 0.447i)20-s − 0.426i·22-s − 1.25i·23-s + (−0.600 − 0.800i)25-s + (0.588 − 0.392i)26-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26757 - 1.11899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26757 - 1.11899i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 6iT - 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 8iT - 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 2iT - 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 2iT - 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 8iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−10.56748351633643151661211230719, −9.396465806080167156949871379253, −8.820292448823425517429833150819, −8.108270293358545525105978610700, −6.53908841650111127574115009615, −5.68693915005834351095918519264, −4.92161691746958411873039800166, −3.99854341562248601697793958572, −2.76524914825219347389641769751, −0.802392914566662689323310059992,
1.95427287501776239266919869683, 3.44721758731866515998227472013, 4.11912991484653771899845166361, 5.53392881829809494019240713335, 6.07866350853521321111553730291, 7.19007111162300088656871640068, 8.211690105870193745802521238935, 9.463905561104978022490565122142, 9.826968328721183710495174574346, 11.06325968634500378071062598839
