| L(s) = 1 | + (0.926 + 0.376i)2-s + (2.90 + 0.651i)3-s + (0.716 + 0.697i)4-s + (−0.0514 − 1.86i)5-s + (2.44 + 1.69i)6-s + (−0.742 − 2.93i)7-s + (0.401 + 0.915i)8-s + (5.30 + 2.50i)9-s + (0.654 − 1.74i)10-s + (0.216 − 2.61i)11-s + (1.62 + 2.49i)12-s + (0.162 + 0.519i)13-s + (0.415 − 2.99i)14-s + (1.06 − 5.45i)15-s + (0.0275 + 0.999i)16-s + (−2.99 + 2.91i)17-s + ⋯ |
| L(s) = 1 | + (0.655 + 0.266i)2-s + (1.67 + 0.375i)3-s + (0.358 + 0.348i)4-s + (−0.0229 − 0.834i)5-s + (0.998 + 0.692i)6-s + (−0.280 − 1.10i)7-s + (0.142 + 0.323i)8-s + (1.76 + 0.834i)9-s + (0.206 − 0.552i)10-s + (0.0652 − 0.787i)11-s + (0.470 + 0.719i)12-s + (0.0450 + 0.144i)13-s + (0.111 − 0.800i)14-s + (0.275 − 1.40i)15-s + (0.00688 + 0.249i)16-s + (−0.726 + 0.707i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.62551 + 0.157793i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.62551 + 0.157793i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.926 - 0.376i)T \) |
| 19 | \( 1 + (0.717 - 4.29i)T \) |
| good | 3 | \( 1 + (-2.90 - 0.651i)T + (2.71 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.0514 + 1.86i)T + (-4.99 + 0.275i)T^{2} \) |
| 7 | \( 1 + (0.742 + 2.93i)T + (-6.15 + 3.33i)T^{2} \) |
| 11 | \( 1 + (-0.216 + 2.61i)T + (-10.8 - 1.81i)T^{2} \) |
| 13 | \( 1 + (-0.162 - 0.519i)T + (-10.6 + 7.40i)T^{2} \) |
| 17 | \( 1 + (2.99 - 2.91i)T + (0.468 - 16.9i)T^{2} \) |
| 23 | \( 1 + (5.32 - 1.19i)T + (20.8 - 9.81i)T^{2} \) |
| 29 | \( 1 + (1.53 + 0.433i)T + (24.7 + 15.1i)T^{2} \) |
| 31 | \( 1 + (0.470 + 0.366i)T + (7.61 + 30.0i)T^{2} \) |
| 37 | \( 1 + (0.574 - 6.93i)T + (-36.4 - 6.08i)T^{2} \) |
| 41 | \( 1 + (-8.71 - 7.58i)T + (5.63 + 40.6i)T^{2} \) |
| 43 | \( 1 + (-0.214 - 0.571i)T + (-32.4 + 28.2i)T^{2} \) |
| 47 | \( 1 + (6.37 + 4.42i)T + (16.4 + 44.0i)T^{2} \) |
| 53 | \( 1 + (2.87 + 1.99i)T + (18.5 + 49.6i)T^{2} \) |
| 59 | \( 1 + (-5.22 - 4.54i)T + (8.10 + 58.4i)T^{2} \) |
| 61 | \( 1 + (3.61 + 4.92i)T + (-18.2 + 58.2i)T^{2} \) |
| 67 | \( 1 + (3.26 + 0.361i)T + (65.3 + 14.6i)T^{2} \) |
| 71 | \( 1 + (4.05 - 5.51i)T + (-21.1 - 67.7i)T^{2} \) |
| 73 | \( 1 + (2.34 - 2.28i)T + (2.01 - 72.9i)T^{2} \) |
| 79 | \( 1 + (-2.66 - 7.10i)T + (-59.5 + 51.8i)T^{2} \) |
| 83 | \( 1 + (-1.96 + 1.06i)T + (45.3 - 69.4i)T^{2} \) |
| 89 | \( 1 + (0.562 + 0.547i)T + (2.45 + 88.9i)T^{2} \) |
| 97 | \( 1 + (13.4 - 1.48i)T + (94.6 - 21.2i)T^{2} \) |
| 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
−10.19705911633074219492259736521, −9.482941242855087480794230716568, −8.313899639439549090786326005831, −8.212530673221006427099741502406, −7.05331205871523936989129224802, −5.94789826441921277388141243250, −4.46833307432126854984118547173, −3.97634150284410774038857341836, −3.10811538663552207290616717111, −1.65002018272778267081874955430,
2.20205138952000057257625854087, 2.53746913412307511142826421356, 3.54001854942652237393496746524, 4.65453663337405809571090691318, 6.08825639098570211104428680899, 7.01925929585207466480483296218, 7.64843709419775501481818190885, 8.924250905639316363968800852983, 9.283404476852053095039343508369, 10.32214805498405860499039311059
