| L(s) = 1 | − i·2-s + 1.24·3-s − 4-s + i·5-s − 1.24i·6-s − 2.49i·7-s + i·8-s − 1.44·9-s + 10-s − 5.80i·11-s − 1.24·12-s − 2.49·14-s + 1.24i·15-s + 16-s − 4.29·17-s + 1.44i·18-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s + 0.719·3-s − 0.5·4-s + 0.447i·5-s − 0.509i·6-s − 0.942i·7-s + 0.353i·8-s − 0.481·9-s + 0.316·10-s − 1.74i·11-s − 0.359·12-s − 0.666·14-s + 0.321i·15-s + 0.250·16-s − 1.04·17-s + 0.340i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.022372304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.022372304\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.24T + 3T^{2} \) |
| 7 | \( 1 + 2.49iT - 7T^{2} \) |
| 11 | \( 1 + 5.80iT - 11T^{2} \) |
| 17 | \( 1 + 4.29T + 17T^{2} \) |
| 19 | \( 1 - 4.04iT - 19T^{2} \) |
| 23 | \( 1 - 3.10T + 23T^{2} \) |
| 29 | \( 1 + 5.60T + 29T^{2} \) |
| 31 | \( 1 + 7.70iT - 31T^{2} \) |
| 37 | \( 1 + 2.67iT - 37T^{2} \) |
| 41 | \( 1 - 12.5iT - 41T^{2} \) |
| 43 | \( 1 + 6.98T + 43T^{2} \) |
| 47 | \( 1 - 3.87iT - 47T^{2} \) |
| 53 | \( 1 + 4.93T + 53T^{2} \) |
| 59 | \( 1 + 10.0iT - 59T^{2} \) |
| 61 | \( 1 + 4.93T + 61T^{2} \) |
| 67 | \( 1 + 8.01iT - 67T^{2} \) |
| 71 | \( 1 + 5.48iT - 71T^{2} \) |
| 73 | \( 1 + 8.67iT - 73T^{2} \) |
| 79 | \( 1 + 1.82T + 79T^{2} \) |
| 83 | \( 1 + 14.6iT - 83T^{2} \) |
| 89 | \( 1 - 0.454iT - 89T^{2} \) |
| 97 | \( 1 - 8.69iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.053947032814513724938258882725, −8.162096335309311002352094542639, −7.72757527550156415202801669413, −6.45124050972052473398736837090, −5.75199065604603974340786544399, −4.45381374644354356313776306548, −3.48213663614615945758403363601, −3.07782563633889286262413860638, −1.84370188465203423078209376130, −0.32900509271534867055579631166,
1.89247721694668571090948578894, 2.75554475256372367224902261102, 4.05455403171601979152171368282, 4.97738829283400457151935576272, 5.54506200644200595838871109511, 6.80993183086566943475838783993, 7.26241604187176458306641854862, 8.354053642791047798460040890481, 8.953714582685596974193134806272, 9.241636962304498981784386518590
