L(s) = 1 | + (2 − 2i)2-s + (−11.0 − 11.0i)3-s − 8i·4-s + (3.15 − 24.8i)5-s − 44.1·6-s + (−2.46 + 2.46i)7-s + (−16 − 16i)8-s + 163. i·9-s + (−43.2 − 55.9i)10-s − 221.·11-s + (−88.3 + 88.3i)12-s + (98.6 + 98.6i)13-s + 9.85i·14-s + (−308. + 239. i)15-s − 64·16-s + (17.4 − 17.4i)17-s + ⋯ |
L(s) = 1 | + (0.5 − 0.5i)2-s + (−1.22 − 1.22i)3-s − 0.5i·4-s + (0.126 − 0.992i)5-s − 1.22·6-s + (−0.0502 + 0.0502i)7-s + (−0.250 − 0.250i)8-s + 2.01i·9-s + (−0.432 − 0.559i)10-s − 1.83·11-s + (−0.613 + 0.613i)12-s + (0.583 + 0.583i)13-s + 0.0502i·14-s + (−1.37 + 1.06i)15-s − 0.250·16-s + (0.0604 − 0.0604i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1952845711\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1952845711\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2 + 2i)T \) |
| 5 | \( 1 + (-3.15 + 24.8i)T \) |
| 23 | \( 1 + (-77.9 - 77.9i)T \) |
good | 3 | \( 1 + (11.0 + 11.0i)T + 81iT^{2} \) |
| 7 | \( 1 + (2.46 - 2.46i)T - 2.40e3iT^{2} \) |
| 11 | \( 1 + 221.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-98.6 - 98.6i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + (-17.4 + 17.4i)T - 8.35e4iT^{2} \) |
| 19 | \( 1 + 413. iT - 1.30e5T^{2} \) |
| 29 | \( 1 + 453. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.07e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + (1.65e3 - 1.65e3i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.55e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (631. + 631. i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 + (-644. + 644. i)T - 4.87e6iT^{2} \) |
| 53 | \( 1 + (-793. - 793. i)T + 7.89e6iT^{2} \) |
| 59 | \( 1 + 2.17e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.37e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-1.21e3 + 1.21e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 + 8.01e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (5.46e3 + 5.46e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 - 5.75e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (7.12e3 + 7.12e3i)T + 4.74e7iT^{2} \) |
| 89 | \( 1 - 302. iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (5.42e3 - 5.42e3i)T - 8.85e7iT^{2} \) |
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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
−11.08896697840691663941537229764, −10.11503743985858344291678621774, −8.615009636539546474401018420400, −7.51521877457061326695675210412, −6.33210062077573996681778184411, −5.39207648312418491665008276827, −4.69338122810104957620189227772, −2.46440214570370731461423946679, −1.17045278361637383266897619800, −0.07122510018623997684111609979,
2.97053895574226688944467584701, 4.09316065319301467526706224655, 5.43198524408729899343496801726, 5.82210533244858272700678127091, 7.08302941681698814221699238990, 8.303677945811586719587904523282, 9.998074231110290249139851528655, 10.48154181314529280458335078299, 11.17067819618181862798172971999, 12.27621473937374393390428089537