| L(s) = 1 | − 2i·5-s − 2i·7-s + 4i·11-s + (−3 − 2i)13-s − 6·17-s + 2i·19-s − 4·23-s + 25-s − 6·29-s + 2i·31-s − 4·35-s − 8i·37-s + 6i·41-s + 4·43-s + 8i·47-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s − 0.755i·7-s + 1.20i·11-s + (−0.832 − 0.554i)13-s − 1.45·17-s + 0.458i·19-s − 0.834·23-s + 0.200·25-s − 1.11·29-s + 0.359i·31-s − 0.676·35-s − 1.31i·37-s + 0.937i·41-s + 0.609·43-s + 1.16i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + (3 + 2i)T \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 6iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 8iT - 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 14iT - 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6iT - 89T^{2} \) |
| 97 | \( 1 + 12iT - 97T^{2} \) |
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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
−8.947086615030263887080745515539, −7.76440096640186285447361573817, −7.42494851245480490656302833471, −6.42204269132684682157980579314, −5.38645716926003710354156963826, −4.51876373753409751902153534296, −4.07584310642582130621342169523, −2.55664207636096127053931491102, −1.49833013967510971932793711274, 0,
2.06805519646484852845393323924, 2.77838441160495111772534318488, 3.79938643644751388869568428137, 4.86332721033446663525706689525, 5.83403943334061435692279578287, 6.53375458259513312763647170135, 7.20558112674778039172295685694, 8.215251931278956167407195435817, 8.952017119313020679849263325568
