L(s) = 1 | − 2i·5-s + 3·9-s − 6i·13-s + 2·17-s + 25-s + 10i·29-s − 2i·37-s − 10·41-s − 6i·45-s − 7·49-s + 14i·53-s + 10i·61-s − 12·65-s + 6·73-s + 9·81-s + ⋯ |
L(s) = 1 | − 0.894i·5-s + 9-s − 1.66i·13-s + 0.485·17-s + 0.200·25-s + 1.85i·29-s − 0.328i·37-s − 1.56·41-s − 0.894i·45-s − 49-s + 1.92i·53-s + 1.28i·61-s − 1.48·65-s + 0.702·73-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.21123 - 0.501708i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.21123 - 0.501708i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 - 3T^{2} \) |
| 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 10iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 10T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14iT - 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 10iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 18T + 97T^{2} \) |
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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
−12.24465143591019829183704169599, −10.76318786599095307120558263935, −10.06255711279168335279332587926, −8.966760340515443645903341695812, −8.033182871671553594656430544981, −7.04783751610773635462797907079, −5.58010149012749333429319409235, −4.72083209570865226203480983593, −3.27909722254998121465231940274, −1.23246026084251524188760195146,
1.97376455261820330387405606640, 3.60827722963180271858468189137, 4.75086858462215078956990844947, 6.41743200731889103929487159669, 6.99447575780113373208701869739, 8.138667714368049024606349572239, 9.511503879403609031118229552499, 10.13053160696446419990753279668, 11.26093909185056800113423925172, 11.95867331713989356615905972730