| L(s) = 1 | + (1.46 − 1.36i)2-s + (−3.25 + 0.490i)3-s + (0.298 − 3.98i)4-s + (0.948 − 2.41i)5-s + (−4.10 + 5.14i)6-s + (−0.473 − 18.5i)7-s + (−4.98 − 6.25i)8-s + (−15.4 + 4.76i)9-s + (−1.89 − 4.83i)10-s + (−15.0 − 4.63i)11-s + (0.983 + 13.1i)12-s + (−14.5 − 63.6i)13-s + (−25.8 − 26.4i)14-s + (−1.90 + 8.32i)15-s + (−15.8 − 2.38i)16-s + (−5.02 + 3.42i)17-s + ⋯ |
| L(s) = 1 | + (0.518 − 0.480i)2-s + (−0.625 + 0.0943i)3-s + (0.0373 − 0.498i)4-s + (0.0848 − 0.216i)5-s + (−0.279 + 0.349i)6-s + (−0.0255 − 0.999i)7-s + (−0.220 − 0.276i)8-s + (−0.572 + 0.176i)9-s + (−0.0599 − 0.152i)10-s + (−0.411 − 0.126i)11-s + (0.0236 + 0.315i)12-s + (−0.309 − 1.35i)13-s + (−0.494 − 0.505i)14-s + (−0.0327 + 0.143i)15-s + (−0.247 − 0.0372i)16-s + (−0.0716 + 0.0488i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.769 + 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.401979 - 1.11453i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.401979 - 1.11453i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.46 + 1.36i)T \) |
| 7 | \( 1 + (0.473 + 18.5i)T \) |
| good | 3 | \( 1 + (3.25 - 0.490i)T + (25.8 - 7.95i)T^{2} \) |
| 5 | \( 1 + (-0.948 + 2.41i)T + (-91.6 - 85.0i)T^{2} \) |
| 11 | \( 1 + (15.0 + 4.63i)T + (1.09e3 + 749. i)T^{2} \) |
| 13 | \( 1 + (14.5 + 63.6i)T + (-1.97e3 + 953. i)T^{2} \) |
| 17 | \( 1 + (5.02 - 3.42i)T + (1.79e3 - 4.57e3i)T^{2} \) |
| 19 | \( 1 + (34.8 + 60.3i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-8.56 - 5.84i)T + (4.44e3 + 1.13e4i)T^{2} \) |
| 29 | \( 1 + (-180. - 87.0i)T + (1.52e4 + 1.90e4i)T^{2} \) |
| 31 | \( 1 + (13.9 - 24.1i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-12.4 - 166. i)T + (-5.00e4 + 7.54e3i)T^{2} \) |
| 41 | \( 1 + (89.6 + 112. i)T + (-1.53e4 + 6.71e4i)T^{2} \) |
| 43 | \( 1 + (-308. + 387. i)T + (-1.76e4 - 7.75e4i)T^{2} \) |
| 47 | \( 1 + (-326. + 303. i)T + (7.75e3 - 1.03e5i)T^{2} \) |
| 53 | \( 1 + (17.3 - 231. i)T + (-1.47e5 - 2.21e4i)T^{2} \) |
| 59 | \( 1 + (81.3 + 207. i)T + (-1.50e5 + 1.39e5i)T^{2} \) |
| 61 | \( 1 + (-32.6 - 435. i)T + (-2.24e5 + 3.38e4i)T^{2} \) |
| 67 | \( 1 + (306. - 530. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (149. - 71.8i)T + (2.23e5 - 2.79e5i)T^{2} \) |
| 73 | \( 1 + (194. + 180. i)T + (2.90e4 + 3.87e5i)T^{2} \) |
| 79 | \( 1 + (528. + 916. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + (-217. + 951. i)T + (-5.15e5 - 2.48e5i)T^{2} \) |
| 89 | \( 1 + (-976. + 301. i)T + (5.82e5 - 3.97e5i)T^{2} \) |
| 97 | \( 1 + 1.05e3T + 9.12e5T^{2} \) |
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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
−13.01698465487107987787563332354, −11.99372102376121592111542935461, −10.71375854346573633341035738003, −10.41009234866097451931397290450, −8.661836748093365009778830371999, −7.16012405959854260609471055943, −5.68115869845125460433397134332, −4.70449292482276055170393701014, −2.99392639621507320830585252358, −0.60864549140943726829541722511,
2.59246032927398408571860680462, 4.58575223517248221757232397990, 5.86170559856575630875164749755, 6.64648374531268363840125365945, 8.220589219256766565231199358588, 9.362067720430041318253508627305, 10.95067750339994477187828678498, 11.97109842552576875414649338072, 12.58275665507408906387451459692, 14.07925980243976774689771882924
