| L(s) = 1 | + 2i·7-s − 2i·13-s + 6i·17-s + 4·19-s + 6i·23-s + 6·29-s − 4·31-s + 2i·37-s − 6·41-s + 10i·43-s + 6i·47-s + 3·49-s − 6i·53-s + 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.755i·7-s − 0.554i·13-s + 1.45i·17-s + 0.917·19-s + 1.25i·23-s + 1.11·29-s − 0.718·31-s + 0.328i·37-s − 0.937·41-s + 1.52i·43-s + 0.875i·47-s + 0.428·49-s − 0.824i·53-s + 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24067 + 0.766779i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24067 + 0.766779i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 2iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−10.15439932560684357266135977229, −9.489725898311464400300868502613, −8.482322343258797703440780631439, −7.907163243023524331780009047822, −6.78989105422016221469085682128, −5.81742469470801246302065654171, −5.16233867670098546519770859243, −3.83796355583813629637042847422, −2.84618538873820592025711437490, −1.47187082659401271260004168201,
0.75253694333488070631697540925, 2.39193764197407511426374929168, 3.60318739957152218051263604951, 4.62082594621544345676391001556, 5.47809821580915633613967569270, 6.84957622951006660848628738644, 7.16773504798249208200477013674, 8.326502679820198834208779532300, 9.156164047109138511474296616246, 10.01570811954058452992744038126
