| L(s) = 1 | + (3.18 + 3.98i)2-s + (−4.68 + 2.25i)3-s + (−4.00 + 17.5i)4-s + (4.76 − 2.29i)5-s + (−23.9 − 11.5i)6-s + (7.38 − 16.9i)7-s + (−46.0 + 22.1i)8-s + (0.0387 − 0.0486i)9-s + (24.3 + 11.7i)10-s + (33.6 + 42.1i)11-s + (−20.8 − 91.3i)12-s + (17.2 + 21.5i)13-s + (91.2 − 24.5i)14-s + (−17.1 + 21.5i)15-s + (−104. − 50.5i)16-s + (−22.3 − 97.7i)17-s + ⋯ |
| L(s) = 1 | + (1.12 + 1.40i)2-s + (−0.902 + 0.434i)3-s + (−0.501 + 2.19i)4-s + (0.426 − 0.205i)5-s + (−1.62 − 0.783i)6-s + (0.398 − 0.917i)7-s + (−2.03 + 0.979i)8-s + (0.00143 − 0.00180i)9-s + (0.768 + 0.370i)10-s + (0.921 + 1.15i)11-s + (−0.501 − 2.19i)12-s + (0.367 + 0.460i)13-s + (1.74 − 0.469i)14-s + (−0.295 + 0.370i)15-s + (−1.63 − 0.789i)16-s + (−0.318 − 1.39i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 - 0.638i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.770 - 0.638i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.636198 + 1.76499i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.636198 + 1.76499i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-7.38 + 16.9i)T \) |
| good | 2 | \( 1 + (-3.18 - 3.98i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (4.68 - 2.25i)T + (16.8 - 21.1i)T^{2} \) |
| 5 | \( 1 + (-4.76 + 2.29i)T + (77.9 - 97.7i)T^{2} \) |
| 11 | \( 1 + (-33.6 - 42.1i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (-17.2 - 21.5i)T + (-488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (22.3 + 97.7i)T + (-4.42e3 + 2.13e3i)T^{2} \) |
| 19 | \( 1 - 70.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + (-32.8 + 143. i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 29 | \( 1 + (17.4 + 76.3i)T + (-2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 + 79.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + (54.6 + 239. i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + (277. - 133. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-327. - 157. i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (-165. - 207. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (43.4 - 190. i)T + (-1.34e5 - 6.45e4i)T^{2} \) |
| 59 | \( 1 + (645. + 310. i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (106. + 466. i)T + (-2.04e5 + 9.84e4i)T^{2} \) |
| 67 | \( 1 - 129.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-102. + 448. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (244. - 306. i)T + (-8.65e4 - 3.79e5i)T^{2} \) |
| 79 | \( 1 - 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + (835. - 1.04e3i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (362. - 454. i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 - 935.T + 9.12e5T^{2} \) |
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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
−15.65251423203101956633272111763, −14.31656031888163826985809811439, −13.73618180173828069178002776980, −12.35124740293901312966344150848, −11.19299976949556731583870430047, −9.411145194560683611771725204685, −7.51977239096173178520035947126, −6.49175908419492148443665079204, −5.09899810439678156101207670289, −4.27889266803393927901293959146,
1.44652433270888279223332610341, 3.43633113482188240826042762474, 5.50673536286794127400745357572, 6.10033314342762533412550432017, 8.962862615928672028524684354529, 10.58098282364070747927082429630, 11.52377348654389441668811334734, 12.08989572307209856341838472253, 13.25845462522440051283113083320, 14.22958757581781810610822577574
