| L(s) = 1 | + (0.284 − 1.97i)2-s + (−1.24 − 2.72i)3-s + (−3.83 − 1.12i)4-s + (0.957 + 1.10i)5-s + (−5.75 + 1.69i)6-s + (−12.8 − 8.25i)7-s + (−3.32 + 7.27i)8-s + (−5.89 + 6.80i)9-s + (2.46 − 1.58i)10-s + (−2.48 − 17.2i)11-s + (1.70 + 11.8i)12-s + (−67.9 + 43.6i)13-s + (−19.9 + 23.0i)14-s + (1.82 − 3.99i)15-s + (13.4 + 8.65i)16-s + (−41.0 + 12.0i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.0856 + 0.0988i)5-s + (−0.391 + 0.115i)6-s + (−0.693 − 0.445i)7-s + (−0.146 + 0.321i)8-s + (−0.218 + 0.251i)9-s + (0.0778 − 0.0500i)10-s + (−0.0681 − 0.474i)11-s + (0.0410 + 0.285i)12-s + (−1.44 + 0.931i)13-s + (−0.381 + 0.440i)14-s + (0.0313 − 0.0687i)15-s + (0.210 + 0.135i)16-s + (−0.584 + 0.171i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.564 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.113468 + 0.215149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.113468 + 0.215149i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.284 + 1.97i)T \) |
| 3 | \( 1 + (1.24 + 2.72i)T \) |
| 23 | \( 1 + (31.9 - 105. i)T \) |
| good | 5 | \( 1 + (-0.957 - 1.10i)T + (-17.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (12.8 + 8.25i)T + (142. + 312. i)T^{2} \) |
| 11 | \( 1 + (2.48 + 17.2i)T + (-1.27e3 + 374. i)T^{2} \) |
| 13 | \( 1 + (67.9 - 43.6i)T + (912. - 1.99e3i)T^{2} \) |
| 17 | \( 1 + (41.0 - 12.0i)T + (4.13e3 - 2.65e3i)T^{2} \) |
| 19 | \( 1 + (-69.0 - 20.2i)T + (5.77e3 + 3.70e3i)T^{2} \) |
| 29 | \( 1 + (214. - 63.0i)T + (2.05e4 - 1.31e4i)T^{2} \) |
| 31 | \( 1 + (24.1 - 52.9i)T + (-1.95e4 - 2.25e4i)T^{2} \) |
| 37 | \( 1 + (-210. + 243. i)T + (-7.20e3 - 5.01e4i)T^{2} \) |
| 41 | \( 1 + (161. + 186. i)T + (-9.80e3 + 6.82e4i)T^{2} \) |
| 43 | \( 1 + (87.7 + 192. i)T + (-5.20e4 + 6.00e4i)T^{2} \) |
| 47 | \( 1 + 438.T + 1.03e5T^{2} \) |
| 53 | \( 1 + (217. + 139. i)T + (6.18e4 + 1.35e5i)T^{2} \) |
| 59 | \( 1 + (-329. + 211. i)T + (8.53e4 - 1.86e5i)T^{2} \) |
| 61 | \( 1 + (36.8 - 80.5i)T + (-1.48e5 - 1.71e5i)T^{2} \) |
| 67 | \( 1 + (-75.7 + 526. i)T + (-2.88e5 - 8.47e4i)T^{2} \) |
| 71 | \( 1 + (-49.3 + 343. i)T + (-3.43e5 - 1.00e5i)T^{2} \) |
| 73 | \( 1 + (-183. - 53.8i)T + (3.27e5 + 2.10e5i)T^{2} \) |
| 79 | \( 1 + (-720. + 462. i)T + (2.04e5 - 4.48e5i)T^{2} \) |
| 83 | \( 1 + (470. - 543. i)T + (-8.13e4 - 5.65e5i)T^{2} \) |
| 89 | \( 1 + (268. + 587. i)T + (-4.61e5 + 5.32e5i)T^{2} \) |
| 97 | \( 1 + (-331. - 382. i)T + (-1.29e5 + 9.03e5i)T^{2} \) |
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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
−12.04450412476205568066039566448, −11.20515979799272243939822196622, −10.02707635001416114199507140234, −9.196662788843141785446162243886, −7.63829593290591261857573739285, −6.58277014741008028724548131602, −5.17708353525696286339522230027, −3.62212313918553528320798316602, −2.05524600190398098867462779132, −0.11405280339369257971388517136,
2.93104360319129362629095025426, 4.64811843916352541156859355357, 5.59239776015552039476078451065, 6.84279979957150119132431253487, 7.996604346714562425824778239125, 9.463441660759689228227771359316, 9.873849330085001532198561690072, 11.40848531544304582008622667069, 12.57693181584790325034219953397, 13.27361130214252272264483396949
