| L(s) = 1 | − 2-s + (0.828 + 1.52i)3-s + 4-s + i·5-s + (−0.828 − 1.52i)6-s + 4.83·7-s − 8-s + (−1.62 + 2.52i)9-s − i·10-s + 2i·11-s + (0.828 + 1.52i)12-s + 1.04i·13-s − 4.83·14-s + (−1.52 + 0.828i)15-s + 16-s − 0.140i·17-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (0.478 + 0.878i)3-s + 0.5·4-s + 0.447i·5-s + (−0.338 − 0.620i)6-s + 1.82·7-s − 0.353·8-s + (−0.542 + 0.840i)9-s − 0.316i·10-s + 0.603i·11-s + (0.239 + 0.439i)12-s + 0.288i·13-s − 1.29·14-s + (−0.392 + 0.213i)15-s + 0.250·16-s − 0.0339i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10892 + 0.947448i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10892 + 0.947448i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (-0.828 - 1.52i)T \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 + (-2.65 + 3.45i)T \) |
| good | 7 | \( 1 - 4.83T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 1.04iT - 13T^{2} \) |
| 17 | \( 1 + 0.140iT - 17T^{2} \) |
| 23 | \( 1 + 6.63iT - 23T^{2} \) |
| 29 | \( 1 - 1.65T + 29T^{2} \) |
| 31 | \( 1 - 3.58iT - 31T^{2} \) |
| 37 | \( 1 - 6.36iT - 37T^{2} \) |
| 41 | \( 1 - 1.18T + 41T^{2} \) |
| 43 | \( 1 + 8.76T + 43T^{2} \) |
| 47 | \( 1 - 4.76iT - 47T^{2} \) |
| 53 | \( 1 + 8.42T + 53T^{2} \) |
| 59 | \( 1 + 0.350T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 13.6iT - 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 2.83T + 73T^{2} \) |
| 79 | \( 1 - 10.0iT - 79T^{2} \) |
| 83 | \( 1 - 10.0iT - 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + 7.67iT - 97T^{2} \) |
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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
−10.87127350543911298469015454753, −10.07125238453636804250302875254, −9.157498763852463652250883700825, −8.319652225932404288137566754283, −7.73926399447256793649280147605, −6.64202173628892170770295626462, −5.06800131601068523271653137885, −4.49409032614249574578578552824, −2.92464577046317464303280717704, −1.77863121993985399373691587974,
1.12889941610199753202969980449, 2.01072518769825544870660576035, 3.56229846452421221385664552167, 5.14677836519444915849622522867, 6.03511402047567736187013107921, 7.50048869787407837973944217309, 7.87001926203158129701566399543, 8.583414452867718351829266488408, 9.374124960199499574223327338562, 10.59347909640732493536167885001
