| L(s) = 1 | + 1.36i·2-s + 0.128·4-s − 2.18i·5-s + 4.27i·7-s + 2.91i·8-s + 2.98·10-s + i·11-s + (−0.922 + 3.48i)13-s − 5.84·14-s − 3.72·16-s − 3.79·17-s − 3.17i·19-s − 0.279i·20-s − 1.36·22-s − 4.94·23-s + ⋯ |
| L(s) = 1 | + 0.967i·2-s + 0.0640·4-s − 0.976i·5-s + 1.61i·7-s + 1.02i·8-s + 0.944·10-s + 0.301i·11-s + (−0.255 + 0.966i)13-s − 1.56·14-s − 0.931·16-s − 0.919·17-s − 0.727i·19-s − 0.0625i·20-s − 0.291·22-s − 1.03·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.409665127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.409665127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (0.922 - 3.48i)T \) |
| good | 2 | \( 1 - 1.36iT - 2T^{2} \) |
| 5 | \( 1 + 2.18iT - 5T^{2} \) |
| 7 | \( 1 - 4.27iT - 7T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 + 3.17iT - 19T^{2} \) |
| 23 | \( 1 + 4.94T + 23T^{2} \) |
| 29 | \( 1 - 7.35T + 29T^{2} \) |
| 31 | \( 1 - 0.0727iT - 31T^{2} \) |
| 37 | \( 1 + 3.66iT - 37T^{2} \) |
| 41 | \( 1 - 4.16iT - 41T^{2} \) |
| 43 | \( 1 + 7.11T + 43T^{2} \) |
| 47 | \( 1 - 11.6iT - 47T^{2} \) |
| 53 | \( 1 + 5.11T + 53T^{2} \) |
| 59 | \( 1 - 5.62iT - 59T^{2} \) |
| 61 | \( 1 + 5.40T + 61T^{2} \) |
| 67 | \( 1 - 10.1iT - 67T^{2} \) |
| 71 | \( 1 + 9.62iT - 71T^{2} \) |
| 73 | \( 1 + 6.44iT - 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 7.80iT - 83T^{2} \) |
| 89 | \( 1 - 2.87iT - 89T^{2} \) |
| 97 | \( 1 - 3.79iT - 97T^{2} \) |
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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
−9.604697836734693277079380144952, −8.907421240969223985897881425302, −8.532368102644513211934697432257, −7.62592578956750993172583518682, −6.51907234133314740271051007264, −6.08751886368544092951605193344, −4.97116548686513093203827532843, −4.60662969047318354073067088175, −2.68830354632462641076668609933, −1.86228910561560480257257929282,
0.54937786155498052689895765767, 1.94170420903009795001210527239, 3.12646999506460298962762905357, 3.67576397224552227441098325292, 4.69150838560491921263168769575, 6.23776982313812996283188358079, 6.82932169045586152503858087431, 7.55347307595746641135030553201, 8.436638296494246298627642602604, 9.961674847536421582761971181135
