| L(s) = 1 | − i·2-s − 4-s + (1.14 + 1.91i)5-s − 1.25i·7-s + i·8-s + (1.91 − 1.14i)10-s − 2·11-s − i·13-s − 1.25·14-s + 16-s − 4.80i·17-s + 5.09·19-s + (−1.14 − 1.91i)20-s + 2i·22-s − 2.58i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (0.512 + 0.858i)5-s − 0.474i·7-s + 0.353i·8-s + (0.607 − 0.362i)10-s − 0.603·11-s − 0.277i·13-s − 0.335·14-s + 0.250·16-s − 1.16i·17-s + 1.16·19-s + (−0.256 − 0.429i)20-s + 0.426i·22-s − 0.538i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.664847460\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.664847460\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.14 - 1.91i)T \) |
| 13 | \( 1 + iT \) |
| good | 7 | \( 1 + 1.25iT - 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 17 | \( 1 + 4.80iT - 17T^{2} \) |
| 19 | \( 1 - 5.09T + 19T^{2} \) |
| 23 | \( 1 + 2.58iT - 23T^{2} \) |
| 29 | \( 1 - 5.09T + 29T^{2} \) |
| 31 | \( 1 - 8.58T + 31T^{2} \) |
| 37 | \( 1 + 7.83iT - 37T^{2} \) |
| 41 | \( 1 - 9.67T + 41T^{2} \) |
| 43 | \( 1 - 10.8iT - 43T^{2} \) |
| 47 | \( 1 - 2.74iT - 47T^{2} \) |
| 53 | \( 1 + 2.58iT - 53T^{2} \) |
| 59 | \( 1 + 5.09T + 59T^{2} \) |
| 61 | \( 1 - 13.6T + 61T^{2} \) |
| 67 | \( 1 + 8.58iT - 67T^{2} \) |
| 71 | \( 1 + 5.38T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 11.0iT - 83T^{2} \) |
| 89 | \( 1 - 5.09T + 89T^{2} \) |
| 97 | \( 1 - 6.26iT - 97T^{2} \) |
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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
−9.883299054161386614430605450174, −9.127749904156201660169781845182, −7.915335474487941758843533105084, −7.27745458196097039874890856319, −6.25671730947102659646049525525, −5.29158511588409829874106976146, −4.34648756712762066733214306443, −3.03828797902457109363099467705, −2.54927562617931576705112955131, −0.900262180629255667876591637477,
1.15958865512380511782563721364, 2.60841754072554068031559434368, 4.04350013553737172658457646786, 5.02008764388490064653119302938, 5.67371981606019095070037610909, 6.42295438841421918170517669064, 7.53712532072101633432305258055, 8.373096246229986748228740549654, 8.870031952468582705866310191816, 9.838168377740724280386757651730
