| L(s) = 1 | + (−0.312 + 1.97i)2-s + (−4.02 + 2.04i)3-s + (0.00704 + 0.00229i)4-s + (4.93 − 0.810i)5-s + (−2.78 − 8.57i)6-s + (3.91 − 3.91i)7-s + (−3.63 + 7.13i)8-s + (6.67 − 9.19i)9-s + (0.0571 + 9.99i)10-s + (−2.73 + 1.98i)11-s + (−0.0330 + 0.00523i)12-s + (−3.13 − 19.8i)13-s + (6.49 + 8.94i)14-s + (−18.1 + 13.3i)15-s + (−12.9 − 9.38i)16-s + (17.7 + 9.02i)17-s + ⋯ |
| L(s) = 1 | + (−0.156 + 0.986i)2-s + (−1.34 + 0.682i)3-s + (0.00176 + 0.000572i)4-s + (0.986 − 0.162i)5-s + (−0.464 − 1.42i)6-s + (0.558 − 0.558i)7-s + (−0.454 + 0.891i)8-s + (0.742 − 1.02i)9-s + (0.00571 + 0.999i)10-s + (−0.248 + 0.180i)11-s + (−0.00275 + 0.000435i)12-s + (−0.241 − 1.52i)13-s + (0.464 + 0.638i)14-s + (−1.21 + 0.891i)15-s + (−0.807 − 0.586i)16-s + (1.04 + 0.530i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.136 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.494270 + 0.567141i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.494270 + 0.567141i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-4.93 + 0.810i)T \) |
| good | 2 | \( 1 + (0.312 - 1.97i)T + (-3.80 - 1.23i)T^{2} \) |
| 3 | \( 1 + (4.02 - 2.04i)T + (5.29 - 7.28i)T^{2} \) |
| 7 | \( 1 + (-3.91 + 3.91i)T - 49iT^{2} \) |
| 11 | \( 1 + (2.73 - 1.98i)T + (37.3 - 115. i)T^{2} \) |
| 13 | \( 1 + (3.13 + 19.8i)T + (-160. + 52.2i)T^{2} \) |
| 17 | \( 1 + (-17.7 - 9.02i)T + (169. + 233. i)T^{2} \) |
| 19 | \( 1 + (5.87 - 1.91i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 + (13.6 + 2.16i)T + (503. + 163. i)T^{2} \) |
| 29 | \( 1 + (29.3 + 9.52i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (0.774 + 2.38i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (33.6 - 5.32i)T + (1.30e3 - 423. i)T^{2} \) |
| 41 | \( 1 + (-29.0 - 21.0i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-23.3 - 23.3i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (19.2 + 37.6i)T + (-1.29e3 + 1.78e3i)T^{2} \) |
| 53 | \( 1 + (73.4 - 37.4i)T + (1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-1.53 + 2.11i)T + (-1.07e3 - 3.31e3i)T^{2} \) |
| 61 | \( 1 + (-21.8 + 15.8i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-67.8 - 34.5i)T + (2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-22.6 + 69.7i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-51.6 - 8.18i)T + (5.06e3 + 1.64e3i)T^{2} \) |
| 79 | \( 1 + (37.4 + 12.1i)T + (5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (22.8 - 44.7i)T + (-4.04e3 - 5.57e3i)T^{2} \) |
| 89 | \( 1 + (41.7 + 57.4i)T + (-2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (1.26 + 2.48i)T + (-5.53e3 + 7.61e3i)T^{2} \) |
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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
−17.29295625807971537709185432295, −16.77901279097029436469153183864, −15.58152926586773706260758626119, −14.41016848360566460328866560127, −12.57144314245464428903341573801, −10.98159131166635095353004679255, −10.01333861795390442885845117091, −7.86519325979508714283963360305, −6.06089592288810378399494501234, −5.19594970885838571033013547476,
1.84533285430714077018680931209, 5.54141734882873054540372143661, 6.83345711223090446186017418201, 9.485546509614340535996700693244, 10.88964050137838133853927003188, 11.76121727974677572219941437083, 12.67466200217122299358718064995, 14.25268908180441249693078254991, 16.24458537924087836675632854497, 17.39234459512347619056049811187
