| L(s) = 1 | − 0.539i·2-s + i·3-s + 1.70·4-s + (2.17 − 0.539i)5-s + 0.539·6-s + i·7-s − 2i·8-s − 9-s + (−0.290 − 1.17i)10-s − 11-s + 1.70i·12-s + 3.17i·13-s + 0.539·14-s + (0.539 + 2.17i)15-s + 2.34·16-s + 2.70i·17-s + ⋯ |
| L(s) = 1 | − 0.381i·2-s + 0.577i·3-s + 0.854·4-s + (0.970 − 0.241i)5-s + 0.220·6-s + 0.377i·7-s − 0.707i·8-s − 0.333·9-s + (−0.0919 − 0.370i)10-s − 0.301·11-s + 0.493i·12-s + 0.879i·13-s + 0.144·14-s + (0.139 + 0.560i)15-s + 0.585·16-s + 0.657i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.426485391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.426485391\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 + (-2.17 + 0.539i)T \) |
| 7 | \( 1 - iT \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + 0.539iT - 2T^{2} \) |
| 13 | \( 1 - 3.17iT - 13T^{2} \) |
| 17 | \( 1 - 2.70iT - 17T^{2} \) |
| 19 | \( 1 - 3.34T + 19T^{2} \) |
| 23 | \( 1 - 1.63iT - 23T^{2} \) |
| 29 | \( 1 + 1.24T + 29T^{2} \) |
| 31 | \( 1 - 5.21T + 31T^{2} \) |
| 37 | \( 1 - 3.17iT - 37T^{2} \) |
| 41 | \( 1 + 0.829T + 41T^{2} \) |
| 43 | \( 1 + 1.24iT - 43T^{2} \) |
| 47 | \( 1 + 4.87iT - 47T^{2} \) |
| 53 | \( 1 + 5.92iT - 53T^{2} \) |
| 59 | \( 1 - 4.46T + 59T^{2} \) |
| 61 | \( 1 + 7.44T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 9.21T + 79T^{2} \) |
| 83 | \( 1 + 15.0iT - 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 4.32iT - 97T^{2} \) |
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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
−10.00922487444609107331054683522, −9.215208573666547336683885471362, −8.396244564550454779217301714380, −7.23818047491517900437377397916, −6.31736669811972925749783791318, −5.65978245500293984101340956216, −4.67171883711606721266436460739, −3.43769306012875907659741150430, −2.44812840088827538575083986144, −1.49582256991069986776967106125,
1.18316338219943226111164851371, 2.44576867813478874526591403358, 3.13200741443580217638743461141, 4.95149349207645265797818126183, 5.78495611652960407100686539286, 6.40706785251794310330327276792, 7.31612843647265672441346349901, 7.77086391034867209581298200943, 8.844712389258752842769251589847, 9.894663651325743291514455009146
