L(s) = 1 | + i·2-s − 4-s − i·7-s − i·8-s − i·13-s + 14-s + 16-s + 3i·17-s − 2·19-s − 3i·23-s + 26-s + i·28-s + 3·29-s − 31-s + i·32-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 0.377i·7-s − 0.353i·8-s − 0.277i·13-s + 0.267·14-s + 0.250·16-s + 0.727i·17-s − 0.458·19-s − 0.625i·23-s + 0.196·26-s + 0.188i·28-s + 0.557·29-s − 0.179·31-s + 0.176i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.333052697\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.333052697\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 7iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 9iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 15iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−8.514491285592630417106541837167, −7.86850025021864003922919082565, −7.11688400372109128256356569296, −6.38263968118976671161944492532, −5.73699273207765516982578236559, −4.78183526035563087960705749112, −4.09276954009409360383077629486, −3.16558000415867748122058097079, −1.88979011876853662396838087607, −0.47440218472797002822762346911,
1.06717930086113082484273531165, 2.22236095699846338261164221695, 2.99458037291442977361376496155, 3.97704346989484432646435547383, 4.78443066126930797220887524463, 5.56546816637444679235070848430, 6.43572334442883342116061027384, 7.31353699695357714789958586425, 8.143699875383429433307235246115, 8.892873197570309396618935235142