| L(s) = 1 | + (−2.60 − 0.136i)2-s + (4.66 + 5.76i)3-s + (−1.21 − 0.127i)4-s + (1.25 + 11.1i)5-s + (−11.3 − 15.6i)6-s + (−1.27 + 18.4i)7-s + (23.7 + 3.75i)8-s + (−5.81 + 27.3i)9-s + (−1.73 − 29.0i)10-s + (−17.0 + 3.61i)11-s + (−4.91 − 7.57i)12-s + (19.8 + 10.0i)13-s + (5.83 − 47.8i)14-s + (−58.1 + 59.0i)15-s + (−51.6 − 10.9i)16-s + (40.0 + 104. i)17-s + ⋯ |
| L(s) = 1 | + (−0.919 − 0.0481i)2-s + (0.897 + 1.10i)3-s + (−0.151 − 0.0159i)4-s + (0.111 + 0.993i)5-s + (−0.772 − 1.06i)6-s + (−0.0687 + 0.997i)7-s + (1.04 + 0.165i)8-s + (−0.215 + 1.01i)9-s + (−0.0549 − 0.919i)10-s + (−0.466 + 0.0991i)11-s + (−0.118 − 0.182i)12-s + (0.422 + 0.215i)13-s + (0.111 − 0.913i)14-s + (−1.00 + 1.01i)15-s + (−0.806 − 0.171i)16-s + (0.571 + 1.48i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.241i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.131284 + 1.07223i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.131284 + 1.07223i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-1.25 - 11.1i)T \) |
| 7 | \( 1 + (1.27 - 18.4i)T \) |
| good | 2 | \( 1 + (2.60 + 0.136i)T + (7.95 + 0.836i)T^{2} \) |
| 3 | \( 1 + (-4.66 - 5.76i)T + (-5.61 + 26.4i)T^{2} \) |
| 11 | \( 1 + (17.0 - 3.61i)T + (1.21e3 - 541. i)T^{2} \) |
| 13 | \( 1 + (-19.8 - 10.0i)T + (1.29e3 + 1.77e3i)T^{2} \) |
| 17 | \( 1 + (-40.0 - 104. i)T + (-3.65e3 + 3.28e3i)T^{2} \) |
| 19 | \( 1 + (10.2 + 97.8i)T + (-6.70e3 + 1.42e3i)T^{2} \) |
| 23 | \( 1 + (-8.15 + 155. i)T + (-1.21e4 - 1.27e3i)T^{2} \) |
| 29 | \( 1 + (6.80 - 9.36i)T + (-7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (10.0 - 22.5i)T + (-1.99e4 - 2.21e4i)T^{2} \) |
| 37 | \( 1 + (271. - 176. i)T + (2.06e4 - 4.62e4i)T^{2} \) |
| 41 | \( 1 + (277. - 90.1i)T + (5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + (-245. - 245. i)T + 7.95e4iT^{2} \) |
| 47 | \( 1 + (181. + 69.7i)T + (7.71e4 + 6.94e4i)T^{2} \) |
| 53 | \( 1 + (-148. + 120. i)T + (3.09e4 - 1.45e5i)T^{2} \) |
| 59 | \( 1 + (-385. + 428. i)T + (-2.14e4 - 2.04e5i)T^{2} \) |
| 61 | \( 1 + (-317. + 285. i)T + (2.37e4 - 2.25e5i)T^{2} \) |
| 67 | \( 1 + (84.7 - 32.5i)T + (2.23e5 - 2.01e5i)T^{2} \) |
| 71 | \( 1 + (-518. - 376. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-137. + 212. i)T + (-1.58e5 - 3.55e5i)T^{2} \) |
| 79 | \( 1 + (-409. - 920. i)T + (-3.29e5 + 3.66e5i)T^{2} \) |
| 83 | \( 1 + (-35.9 + 226. i)T + (-5.43e5 - 1.76e5i)T^{2} \) |
| 89 | \( 1 + (-378. - 420. i)T + (-7.36e4 + 7.01e5i)T^{2} \) |
| 97 | \( 1 + (20.7 + 130. i)T + (-8.68e5 + 2.82e5i)T^{2} \) |
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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
−12.79507533406452623107332034443, −11.15920009167622310440690752518, −10.34413590941867189910880510356, −9.727895788505246124094809420169, −8.677850468359621804240996070301, −8.215778616989790604675524930098, −6.57627248022419153085042804013, −4.95171805430968209035831194931, −3.56093706251182781459173529685, −2.28786062489259312303652092415,
0.63146374273455866376030509221, 1.65094424928450272327293818374, 3.68684477429543834095729117957, 5.30617485583229123275079388486, 7.24320216747282874719026936907, 7.75356715565368187793500974625, 8.597324243891048859550689084446, 9.491129283483341147616977890575, 10.43782008178706209906830241022, 11.99144495276171941143667554200
