L(s) = 1 | + (0.258 − 0.965i)3-s + (0.866 − 0.5i)5-s + (−1.67 − 0.965i)7-s + (−0.866 − 0.499i)9-s + (−0.258 − 0.965i)15-s + (−1.36 + 1.36i)21-s + (0.258 + 0.448i)23-s + (0.499 − 0.866i)25-s + (−0.707 + 0.707i)27-s + (−1.5 − 0.866i)29-s − 1.93·35-s + (0.866 − 0.5i)41-s + (1.22 + 0.707i)43-s − 45-s + (0.965 − 1.67i)47-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)3-s + (0.866 − 0.5i)5-s + (−1.67 − 0.965i)7-s + (−0.866 − 0.499i)9-s + (−0.258 − 0.965i)15-s + (−1.36 + 1.36i)21-s + (0.258 + 0.448i)23-s + (0.499 − 0.866i)25-s + (−0.707 + 0.707i)27-s + (−1.5 − 0.866i)29-s − 1.93·35-s + (0.866 − 0.5i)41-s + (1.22 + 0.707i)43-s − 45-s + (0.965 − 1.67i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.009998171\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.009998171\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.258 + 0.965i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
good | 7 | \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.965 + 1.67i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + 1.73iT - T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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
−9.367432426058290177820669217320, −8.828330402671414766740953393872, −7.55141499164128436458963486955, −7.09456604592271074832058776497, −6.10934655610387730494938468986, −5.72332067329311014745171409718, −4.16371728020370826332622017686, −3.19814834111105719778202090707, −2.15355109339415842354511625985, −0.77434470585115840678542513468,
2.33823166073465499462402561241, 2.99682199180946684505409853231, 3.84233585379498582247354340656, 5.20558638680078100139719807957, 5.89743915031330638244984415874, 6.49215325626714137311195338876, 7.58626575487523376451939704978, 8.999821434996974316059061531390, 9.193941114236131850617740382140, 9.835588459437010038800086617425