| L(s) = 1 | + (−1.41 + 4.36i)2-s + (−7.07 − 5.13i)3-s + (−10.5 − 7.69i)4-s + (1.19 + 11.1i)5-s + (32.4 − 23.5i)6-s − 20.8·7-s + (18.8 − 13.7i)8-s + (15.2 + 46.9i)9-s + (−50.2 − 10.5i)10-s + (−2.21 + 6.80i)11-s + (35.3 + 108. i)12-s + (9.36 + 28.8i)13-s + (29.6 − 91.2i)14-s + (48.7 − 84.7i)15-s + (0.784 + 2.41i)16-s + (22.6 − 16.4i)17-s + ⋯ |
| L(s) = 1 | + (−0.501 + 1.54i)2-s + (−1.36 − 0.988i)3-s + (−1.32 − 0.961i)4-s + (0.106 + 0.994i)5-s + (2.20 − 1.60i)6-s − 1.12·7-s + (0.834 − 0.606i)8-s + (0.565 + 1.74i)9-s + (−1.58 − 0.334i)10-s + (−0.0606 + 0.186i)11-s + (0.850 + 2.61i)12-s + (0.199 + 0.614i)13-s + (0.565 − 1.74i)14-s + (0.838 − 1.45i)15-s + (0.0122 + 0.0377i)16-s + (0.323 − 0.235i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0580426 - 0.245305i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0580426 - 0.245305i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.19 - 11.1i)T \) |
| good | 2 | \( 1 + (1.41 - 4.36i)T + (-6.47 - 4.70i)T^{2} \) |
| 3 | \( 1 + (7.07 + 5.13i)T + (8.34 + 25.6i)T^{2} \) |
| 7 | \( 1 + 20.8T + 343T^{2} \) |
| 11 | \( 1 + (2.21 - 6.80i)T + (-1.07e3 - 782. i)T^{2} \) |
| 13 | \( 1 + (-9.36 - 28.8i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (-22.6 + 16.4i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (9.78 - 7.11i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + (16.1 - 49.6i)T + (-9.84e3 - 7.15e3i)T^{2} \) |
| 29 | \( 1 + (119. + 86.6i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (172. - 125. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (22.0 + 67.7i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-159. - 489. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 176.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-181. - 132. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (292. + 212. i)T + (4.60e4 + 1.41e5i)T^{2} \) |
| 59 | \( 1 + (-181. - 558. i)T + (-1.66e5 + 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-244. + 752. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 + (-347. + 252. i)T + (9.29e4 - 2.86e5i)T^{2} \) |
| 71 | \( 1 + (-321. - 233. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (321. - 988. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (989. + 718. i)T + (1.52e5 + 4.68e5i)T^{2} \) |
| 83 | \( 1 + (189. - 137. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + (56.9 - 175. i)T + (-5.70e5 - 4.14e5i)T^{2} \) |
| 97 | \( 1 + (67.1 + 48.7i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−17.67158497540825665846196048460, −16.67068994428179883829591646568, −15.81507977140271153653547813700, −14.27749399918871197566554171089, −12.91843619646215062747168043463, −11.34025805751629443098210472749, −9.704182895486841665632433507173, −7.47210811626473031332049627416, −6.63139599995692454773848345309, −5.79165619994272303217783695209,
0.34553308882335207572775296274, 3.88795072494938541018916164273, 5.70211756809943904690102366779, 9.070249322225668690008029999473, 10.02395812478468856301150026051, 10.97580507988493638265071631392, 12.25797847424978900860941058546, 12.95928033300262610614079400771, 15.73068852254042647885290569591, 16.66852457161502462103220212463
