| L(s) = 1 | + (−4.61 − 0.403i)2-s + (−4.94 − 1.60i)3-s + (13.2 + 2.33i)4-s + (6.75 + 8.91i)5-s + (22.1 + 9.41i)6-s + (−24.2 − 16.9i)7-s + (−24.4 − 6.54i)8-s + (21.8 + 15.8i)9-s + (−27.5 − 43.8i)10-s + (17.9 − 49.1i)11-s + (−61.7 − 32.8i)12-s + (−2.35 − 26.9i)13-s + (105. + 88.1i)14-s + (−19.0 − 54.8i)15-s + (8.91 + 3.24i)16-s + (−5.36 + 1.43i)17-s + ⋯ |
| L(s) = 1 | + (−1.63 − 0.142i)2-s + (−0.950 − 0.309i)3-s + (1.65 + 0.292i)4-s + (0.603 + 0.797i)5-s + (1.50 + 0.640i)6-s + (−1.30 − 0.917i)7-s + (−1.07 − 0.289i)8-s + (0.808 + 0.588i)9-s + (−0.871 − 1.38i)10-s + (0.490 − 1.34i)11-s + (−1.48 − 0.790i)12-s + (−0.0503 − 0.575i)13-s + (2.00 + 1.68i)14-s + (−0.327 − 0.944i)15-s + (0.139 + 0.0506i)16-s + (−0.0765 + 0.0205i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.254 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.107223 + 0.139048i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.107223 + 0.139048i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (4.94 + 1.60i)T \) |
| 5 | \( 1 + (-6.75 - 8.91i)T \) |
| good | 2 | \( 1 + (4.61 + 0.403i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (24.2 + 16.9i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (-17.9 + 49.1i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (2.35 + 26.9i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (5.36 - 1.43i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (51.6 - 29.8i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-106. - 151. i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (108. - 90.8i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (-3.14 + 17.8i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-37.0 - 138. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (11.4 - 13.6i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (134. + 289. i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (265. - 378. i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (226. - 226. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (619. - 225. i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (-133. - 756. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-173. + 15.1i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (-188. - 108. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-38.4 + 143. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (306. + 364. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (18.4 - 210. i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (253. + 439. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.62e3 - 757. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
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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
−12.99416582977716457001655630104, −11.43123401717414582642794836392, −10.74051110845765367813798423322, −10.09930853560118949965168117410, −9.164956597873780879855718759653, −7.58393615548326197748017188411, −6.74597110868850268995089999634, −5.93305485262416550164295902293, −3.25467997879042236838427494503, −1.20097339324695381669962089081,
0.18957080131333946625930397593, 1.96830589732119736988244216005, 4.69451150471742149657196019899, 6.27637779983065567296243373430, 6.86354927107445437464996918520, 8.689578710201447169622571015879, 9.581874560791442652123290315037, 9.809443192236646218803829811364, 11.15987432059666857179682222188, 12.36885339840434765733496359304
