| L(s) = 1 | − 1.19·2-s − 0.567·4-s − 0.563·5-s − 0.343·7-s + 3.07·8-s + 0.674·10-s − 3.12·11-s + 4.44·13-s + 0.411·14-s − 2.54·16-s − 7.44·17-s + 3.15·19-s + 0.319·20-s + 3.73·22-s + 3.69·23-s − 4.68·25-s − 5.32·26-s + 0.195·28-s − 29-s + 9.66·31-s − 3.10·32-s + 8.91·34-s + 0.193·35-s + 9.34·37-s − 3.77·38-s − 1.73·40-s + 8.00·41-s + ⋯ |
| L(s) = 1 | − 0.846·2-s − 0.283·4-s − 0.251·5-s − 0.129·7-s + 1.08·8-s + 0.213·10-s − 0.941·11-s + 1.23·13-s + 0.109·14-s − 0.635·16-s − 1.80·17-s + 0.724·19-s + 0.0714·20-s + 0.796·22-s + 0.771·23-s − 0.936·25-s − 1.04·26-s + 0.0368·28-s − 0.185·29-s + 1.73·31-s − 0.548·32-s + 1.52·34-s + 0.0327·35-s + 1.53·37-s − 0.612·38-s − 0.273·40-s + 1.25·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7372454632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7372454632\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 1.19T + 2T^{2} \) |
| 5 | \( 1 + 0.563T + 5T^{2} \) |
| 7 | \( 1 + 0.343T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 - 4.44T + 13T^{2} \) |
| 17 | \( 1 + 7.44T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 - 3.69T + 23T^{2} \) |
| 31 | \( 1 - 9.66T + 31T^{2} \) |
| 37 | \( 1 - 9.34T + 37T^{2} \) |
| 41 | \( 1 - 8.00T + 41T^{2} \) |
| 43 | \( 1 + 0.957T + 43T^{2} \) |
| 47 | \( 1 - 2.49T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 6.40T + 59T^{2} \) |
| 61 | \( 1 - 13.9T + 61T^{2} \) |
| 67 | \( 1 + 11.0T + 67T^{2} \) |
| 71 | \( 1 - 9.26T + 71T^{2} \) |
| 73 | \( 1 - 8.09T + 73T^{2} \) |
| 79 | \( 1 + 8.37T + 79T^{2} \) |
| 83 | \( 1 - 3.27T + 83T^{2} \) |
| 89 | \( 1 + 5.67T + 89T^{2} \) |
| 97 | \( 1 - 6.01T + 97T^{2} \) |
| 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
−10.16970606459244773127765825656, −9.386260699079276531851095843830, −8.568326336690874613344928544397, −8.003325930514568014976500267977, −7.06042309665700392607756385026, −5.98751748541057374276020733927, −4.79860764282221541789026776097, −3.93273872588995248953549851087, −2.45485417185104925606707125232, −0.810839385700066633029461873531,
0.810839385700066633029461873531, 2.45485417185104925606707125232, 3.93273872588995248953549851087, 4.79860764282221541789026776097, 5.98751748541057374276020733927, 7.06042309665700392607756385026, 8.003325930514568014976500267977, 8.568326336690874613344928544397, 9.386260699079276531851095843830, 10.16970606459244773127765825656
