L(s) = 1 | + (−0.258 + 0.965i)2-s + (0.707 − 0.707i)3-s + (0.500 + 0.866i)6-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (0.965 + 0.258i)13-s + (−0.866 + 0.500i)14-s + (−0.5 − 0.866i)16-s + (−0.258 + 0.965i)17-s + (0.965 + 0.258i)18-s + (−0.866 − 0.5i)19-s + 1.00·21-s + 0.999i·24-s + (−0.499 + 0.866i)26-s + (−0.707 − 0.707i)27-s + ⋯ |
L(s) = 1 | + (−0.258 + 0.965i)2-s + (0.707 − 0.707i)3-s + (0.500 + 0.866i)6-s + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (0.965 + 0.258i)13-s + (−0.866 + 0.500i)14-s + (−0.5 − 0.866i)16-s + (−0.258 + 0.965i)17-s + (0.965 + 0.258i)18-s + (−0.866 − 0.5i)19-s + 1.00·21-s + 0.999i·24-s + (−0.499 + 0.866i)26-s + (−0.707 − 0.707i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.410098365\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.410098365\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 2 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - iT^{2} \) |
| 29 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.258 - 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \) |
| 53 | \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + 2T + T^{2} \) |
| 73 | \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (0.258 + 0.965i)T + (-0.866 + 0.5i)T^{2} \) |
| 89 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (0.965 - 0.258i)T + (0.866 - 0.5i)T^{2} \) |
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
−9.189164220187618661665505634636, −8.583226305235080019589441328021, −8.296591136709525968916893599602, −7.47487465714664736637391474876, −6.45369490331203329685257398117, −6.21907595265765632459265590765, −5.03956849871514225659937931066, −3.77266632992031315665667382987, −2.59423705567609324146549979766, −1.71710089531194582892474345443,
1.31296636736101174825138448459, 2.40365667404626328749019836380, 3.35988578309785708866730364071, 4.14272414368454757616696711833, 5.02183334435202348134933835355, 6.25812986912009067542057710429, 7.21595901499485996283343328513, 8.250527551430187550985831276307, 8.756896816610440773430961172423, 9.674533377327000463156271679046