| L(s) = 1 | + (−0.445 + 1.94i)2-s + (−3.63 + 4.55i)3-s + (−3.60 − 1.73i)4-s + (12.3 − 15.4i)5-s + (−7.26 − 9.10i)6-s + (−4.74 − 17.9i)7-s + (4.98 − 6.25i)8-s + (−1.53 − 6.73i)9-s + (24.6 + 30.9i)10-s + (7.76 − 34.0i)11-s + (20.9 − 10.1i)12-s + (−7.08 + 31.0i)13-s + (37.0 − 1.28i)14-s + (25.6 + 112. i)15-s + (9.97 + 12.5i)16-s + (103. − 49.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.157 + 0.689i)2-s + (−0.698 + 0.876i)3-s + (−0.450 − 0.216i)4-s + (1.10 − 1.38i)5-s + (−0.494 − 0.619i)6-s + (−0.256 − 0.966i)7-s + (0.220 − 0.276i)8-s + (−0.0569 − 0.249i)9-s + (0.779 + 0.977i)10-s + (0.212 − 0.932i)11-s + (0.504 − 0.243i)12-s + (−0.151 + 0.662i)13-s + (0.706 − 0.0245i)14-s + (0.440 + 1.93i)15-s + (0.155 + 0.195i)16-s + (1.47 − 0.708i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.20132 - 0.0738254i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20132 - 0.0738254i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.445 - 1.94i)T \) |
| 7 | \( 1 + (4.74 + 17.9i)T \) |
| good | 3 | \( 1 + (3.63 - 4.55i)T + (-6.00 - 26.3i)T^{2} \) |
| 5 | \( 1 + (-12.3 + 15.4i)T + (-27.8 - 121. i)T^{2} \) |
| 11 | \( 1 + (-7.76 + 34.0i)T + (-1.19e3 - 577. i)T^{2} \) |
| 13 | \( 1 + (7.08 - 31.0i)T + (-1.97e3 - 953. i)T^{2} \) |
| 17 | \( 1 + (-103. + 49.6i)T + (3.06e3 - 3.84e3i)T^{2} \) |
| 19 | \( 1 - 3.70T + 6.85e3T^{2} \) |
| 23 | \( 1 + (34.2 + 16.4i)T + (7.58e3 + 9.51e3i)T^{2} \) |
| 29 | \( 1 + (-48.4 + 23.3i)T + (1.52e4 - 1.90e4i)T^{2} \) |
| 31 | \( 1 + 67.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-397. + 191. i)T + (3.15e4 - 3.96e4i)T^{2} \) |
| 41 | \( 1 + (-107. + 135. i)T + (-1.53e4 - 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-29.6 - 37.1i)T + (-1.76e4 + 7.75e4i)T^{2} \) |
| 47 | \( 1 + (73.6 - 322. i)T + (-9.35e4 - 4.50e4i)T^{2} \) |
| 53 | \( 1 + (197. + 95.0i)T + (9.28e4 + 1.16e5i)T^{2} \) |
| 59 | \( 1 + (170. + 214. i)T + (-4.57e4 + 2.00e5i)T^{2} \) |
| 61 | \( 1 + (646. - 311. i)T + (1.41e5 - 1.77e5i)T^{2} \) |
| 67 | \( 1 - 960.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-231. - 111. i)T + (2.23e5 + 2.79e5i)T^{2} \) |
| 73 | \( 1 + (90.6 + 397. i)T + (-3.50e5 + 1.68e5i)T^{2} \) |
| 79 | \( 1 + 746.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-288. - 1.26e3i)T + (-5.15e5 + 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-310. - 1.36e3i)T + (-6.35e5 + 3.05e5i)T^{2} \) |
| 97 | \( 1 + 525.T + 9.12e5T^{2} \) |
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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
−13.67412811511809753974869641949, −12.51597009963145519994630581249, −11.04055529065312506500833419010, −9.810811576646054317129793219030, −9.361438891196104271636880062898, −7.88298715286490262981952095083, −6.14223433921573466352256881995, −5.27736424540089831475516409794, −4.23401337797205200474640665603, −0.864448824988190512586610150520,
1.71953905444482540197808877008, 3.02403725694417271924046952901, 5.62136675537432314039130073853, 6.41621455277052099016630090408, 7.68206653923771549214754135846, 9.570184359530981022649194894064, 10.21139292387076770161218811901, 11.43987318771319342732694775887, 12.38750138742479778334027589144, 13.06453305768760511316213897773
