| L(s) = 1 | + (1.30 − 1.14i)3-s − 3.53i·5-s + 3.09i·7-s + (0.383 − 2.97i)9-s − 5.60·11-s + 0.378·13-s + (−4.04 − 4.59i)15-s + 1.90i·17-s + i·19-s + (3.54 + 4.02i)21-s − 7.91·23-s − 7.48·25-s + (−2.90 − 4.30i)27-s − 9.18i·29-s − 6.82i·31-s + ⋯ |
| L(s) = 1 | + (0.750 − 0.660i)3-s − 1.58i·5-s + 1.17i·7-s + (0.127 − 0.991i)9-s − 1.69·11-s + 0.105·13-s + (−1.04 − 1.18i)15-s + 0.463i·17-s + 0.229i·19-s + (0.772 + 0.878i)21-s − 1.64·23-s − 1.49·25-s + (−0.559 − 0.829i)27-s − 1.70i·29-s − 1.22i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1824 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.003505337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.003505337\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.30 + 1.14i)T \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 + 3.53iT - 5T^{2} \) |
| 7 | \( 1 - 3.09iT - 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 - 0.378T + 13T^{2} \) |
| 17 | \( 1 - 1.90iT - 17T^{2} \) |
| 23 | \( 1 + 7.91T + 23T^{2} \) |
| 29 | \( 1 + 9.18iT - 29T^{2} \) |
| 31 | \( 1 + 6.82iT - 31T^{2} \) |
| 37 | \( 1 + 2.82T + 37T^{2} \) |
| 41 | \( 1 - 0.521iT - 41T^{2} \) |
| 43 | \( 1 - 6.60iT - 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 + 8.42iT - 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + 4.72T + 61T^{2} \) |
| 67 | \( 1 + 9.11iT - 67T^{2} \) |
| 71 | \( 1 - 1.69T + 71T^{2} \) |
| 73 | \( 1 + 2.30T + 73T^{2} \) |
| 79 | \( 1 - 5.74iT - 79T^{2} \) |
| 83 | \( 1 - 8.83T + 83T^{2} \) |
| 89 | \( 1 - 13.1iT - 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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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
−8.595519988308601264035593741596, −8.028317645991688376607081362566, −7.901748938903643633134524179777, −6.24242781990522866349747659253, −5.68629771349235996095555519857, −4.83273181775813261140822830213, −3.79990836041447912710540227406, −2.46105588754735542602416848448, −1.87067910407148757270171437083, −0.30064156802148288787926834841,
2.07004128890514300794244761039, 3.05219911690984251144250914860, 3.53855433208543009007520722425, 4.62634294341667342990328421914, 5.52980588684303518529192122834, 6.80302744652283305149786745522, 7.36093280138969528834625965658, 7.933174548901845049045476222336, 8.847486775521406206839454186758, 10.10037033785201394771717223652
