| L(s) = 1 | − 1.25·2-s − 1.78·3-s − 0.414·4-s + 5-s + 2.24·6-s − 0.828·7-s + 3.04·8-s + 0.171·9-s − 1.25·10-s + 2·11-s + 0.737·12-s − 4.29·13-s + 1.04·14-s − 1.78·15-s − 3·16-s + 7.65·17-s − 0.216·18-s − 0.414·20-s + 1.47·21-s − 2.51·22-s − 0.828·23-s − 5.41·24-s + 25-s + 5.41·26-s + 5.03·27-s + 0.343·28-s − 8.59·29-s + ⋯ |
| L(s) = 1 | − 0.890·2-s − 1.02·3-s − 0.207·4-s + 0.447·5-s + 0.915·6-s − 0.313·7-s + 1.07·8-s + 0.0571·9-s − 0.398·10-s + 0.603·11-s + 0.212·12-s − 1.19·13-s + 0.278·14-s − 0.459·15-s − 0.750·16-s + 1.85·17-s − 0.0509·18-s − 0.0926·20-s + 0.321·21-s − 0.536·22-s − 0.172·23-s − 1.10·24-s + 0.200·25-s + 1.06·26-s + 0.969·27-s + 0.0648·28-s − 1.59·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5201961364\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5201961364\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + 1.25T + 2T^{2} \) |
| 3 | \( 1 + 1.78T + 3T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 4.29T + 13T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 + 8.59T + 29T^{2} \) |
| 31 | \( 1 + 3.56T + 31T^{2} \) |
| 37 | \( 1 + 0.737T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 7.86T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 + 2.82T + 61T^{2} \) |
| 67 | \( 1 + 6.81T + 67T^{2} \) |
| 71 | \( 1 + 13.6T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 7.12T + 79T^{2} \) |
| 83 | \( 1 - 8.82T + 83T^{2} \) |
| 89 | \( 1 - 15.7T + 89T^{2} \) |
| 97 | \( 1 + 4.29T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.369491778227454632504485444292, −8.703020067506486367481499145146, −7.56659948305816631947221108638, −7.13641516447116453110366122203, −5.89697178596742466041060977826, −5.44551906650689574645642644387, −4.51986435273402998445167634174, −3.32461300071538811162404986123, −1.80536949849057093925698171919, −0.60122646125341031000255316467,
0.60122646125341031000255316467, 1.80536949849057093925698171919, 3.32461300071538811162404986123, 4.51986435273402998445167634174, 5.44551906650689574645642644387, 5.89697178596742466041060977826, 7.13641516447116453110366122203, 7.56659948305816631947221108638, 8.703020067506486367481499145146, 9.369491778227454632504485444292
