L(s) = 1 | − i·3-s + 2·9-s + 2·11-s − 5i·13-s + 4i·17-s + 2·19-s − i·23-s − 5i·27-s + 3·29-s + 7·31-s − 2i·33-s + 2i·37-s − 5·39-s − 9·41-s − 4i·43-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.666·9-s + 0.603·11-s − 1.38i·13-s + 0.970i·17-s + 0.458·19-s − 0.208i·23-s − 0.962i·27-s + 0.557·29-s + 1.25·31-s − 0.348i·33-s + 0.328i·37-s − 0.800·39-s − 1.40·41-s − 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.239291412\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.239291412\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + iT \) |
good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 5iT - 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 9T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 + T + 71T^{2} \) |
| 73 | \( 1 - iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 - 4iT - 97T^{2} \) |
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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
−8.179601580147223344006452475914, −7.49242049364059786024122014112, −6.71432553154183620787476990508, −6.16947130669112473693139586873, −5.30802100754912077238355864120, −4.43446087846467611734654205892, −3.59259555539317274659978741662, −2.70129889536784819637221992176, −1.59101803780633745754875348953, −0.76274912844162435656476452823,
1.04584270128031457834865558683, 2.07597613751231664573489242707, 3.20122168115195979571617579194, 4.05583855104512051400844996245, 4.62941543333005518116565524367, 5.32107833308116842769210031163, 6.47325099482051362561726648766, 6.90025614703684659844922500466, 7.63038858744467132932930325995, 8.659274547511491376797276220149