| L(s) = 1 | + (2.22 − 0.721i)2-s + (−9.09 − 12.5i)3-s + (−21.4 + 15.6i)4-s + (−55.6 + 4.76i)5-s + (−29.2 − 21.2i)6-s + 43.5i·7-s + (−80.3 + 110. i)8-s + (1.12 − 3.45i)9-s + (−120. + 50.7i)10-s + (−175. − 539. i)11-s + (390. + 126. i)12-s + (−283. − 92.1i)13-s + (31.4 + 96.7i)14-s + (566. + 653. i)15-s + (163. − 504. i)16-s + (−404. + 556. i)17-s + ⋯ |
| L(s) = 1 | + (0.392 − 0.127i)2-s + (−0.583 − 0.802i)3-s + (−0.671 + 0.487i)4-s + (−0.996 + 0.0851i)5-s + (−0.331 − 0.240i)6-s + 0.335i·7-s + (−0.443 + 0.611i)8-s + (0.00461 − 0.0142i)9-s + (−0.380 + 0.160i)10-s + (−0.436 − 1.34i)11-s + (0.783 + 0.254i)12-s + (−0.465 − 0.151i)13-s + (0.0428 + 0.131i)14-s + (0.649 + 0.750i)15-s + (0.159 − 0.492i)16-s + (−0.339 + 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0806i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.00884859 + 0.219036i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00884859 + 0.219036i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (55.6 - 4.76i)T \) |
| good | 2 | \( 1 + (-2.22 + 0.721i)T + (25.8 - 18.8i)T^{2} \) |
| 3 | \( 1 + (9.09 + 12.5i)T + (-75.0 + 231. i)T^{2} \) |
| 7 | \( 1 - 43.5iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (175. + 539. i)T + (-1.30e5 + 9.46e4i)T^{2} \) |
| 13 | \( 1 + (283. + 92.1i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (404. - 556. i)T + (-4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-912. - 663. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + (-19.1 + 6.20i)T + (5.20e6 - 3.78e6i)T^{2} \) |
| 29 | \( 1 + (5.68e3 - 4.13e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (997. + 724. i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (1.20e4 + 3.92e3i)T + (5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-5.77e3 + 1.77e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 1.44e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-3.65e3 - 5.03e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.23e4 + 1.69e4i)T + (-1.29e8 + 3.97e8i)T^{2} \) |
| 59 | \( 1 + (-1.01e4 + 3.13e4i)T + (-5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (3.78e3 + 1.16e4i)T + (-6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 + (3.55e4 - 4.88e4i)T + (-4.17e8 - 1.28e9i)T^{2} \) |
| 71 | \( 1 + (1.41e4 - 1.02e4i)T + (5.57e8 - 1.71e9i)T^{2} \) |
| 73 | \( 1 + (-1.97e4 + 6.43e3i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (5.17e4 - 3.75e4i)T + (9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (-4.91e4 + 6.76e4i)T + (-1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 + (3.87e4 + 1.19e5i)T + (-4.51e9 + 3.28e9i)T^{2} \) |
| 97 | \( 1 + (-3.87e4 - 5.33e4i)T + (-2.65e9 + 8.16e9i)T^{2} \) |
| 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
−15.97636761469435956084776672123, −14.45321086039992044771307084780, −13.01616586138306402819762653156, −12.20736869823765824705428916397, −11.16518164987775312294672542062, −8.765407860471369237760296474397, −7.47509479267269155680177678890, −5.55902674081827311861910914764, −3.53184511553169946838086457582, −0.14133218186573710357391492505,
4.22651034330342771256982619455, 5.10298835839231057101787578561, 7.36870564319467898493617075352, 9.439169622555459796182972556486, 10.57926777083068836539287879495, 12.02465973767375917293713691910, 13.43887049649749030698859887990, 15.03181149906796707985899827606, 15.65505878381416198570299840206, 16.98918951663329630289729830515
