| L(s) = 1 | + (0.951 − 0.309i)2-s + (−1.47 + 0.910i)3-s + (0.809 − 0.587i)4-s + (0.866 − 0.5i)5-s + (−1.11 + 1.32i)6-s + (0.140 + 1.34i)7-s + (0.587 − 0.809i)8-s + (1.34 − 2.68i)9-s + (0.669 − 0.743i)10-s + (−3.37 − 1.50i)11-s + (−0.656 + 1.60i)12-s + (−1.36 − 6.41i)13-s + (0.548 + 1.23i)14-s + (−0.820 + 1.52i)15-s + (0.309 − 0.951i)16-s + (5.12 − 2.28i)17-s + ⋯ |
| L(s) = 1 | + (0.672 − 0.218i)2-s + (−0.850 + 0.525i)3-s + (0.404 − 0.293i)4-s + (0.387 − 0.223i)5-s + (−0.457 + 0.539i)6-s + (0.0532 + 0.506i)7-s + (0.207 − 0.286i)8-s + (0.446 − 0.894i)9-s + (0.211 − 0.235i)10-s + (−1.01 − 0.452i)11-s + (−0.189 + 0.462i)12-s + (−0.377 − 1.77i)13-s + (0.146 + 0.329i)14-s + (−0.211 + 0.393i)15-s + (0.0772 − 0.237i)16-s + (1.24 − 0.553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.459 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.46230 - 0.889998i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46230 - 0.889998i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.951 + 0.309i)T \) |
| 3 | \( 1 + (1.47 - 0.910i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 + (-4.68 + 3.01i)T \) |
| good | 7 | \( 1 + (-0.140 - 1.34i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (3.37 + 1.50i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (1.36 + 6.41i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-5.12 + 2.28i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (1.80 + 0.383i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-4.04 - 2.93i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-0.419 - 1.29i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (3.11 + 1.79i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.81 - 1.63i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.02 + 9.52i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (2.26 + 0.737i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.959 + 9.13i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (7.31 - 6.58i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 1.70iT - 61T^{2} \) |
| 67 | \( 1 + (0.0499 + 0.0865i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.04 - 0.845i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-1.50 + 3.37i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-0.331 - 0.745i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (3.07 - 3.41i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (9.31 - 6.77i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-13.8 + 10.0i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.26265713091741870497330943403, −9.381330830685600241362338632517, −8.202670525611060164450738048682, −7.24794627004258520063002896243, −5.98083989676737870898664516285, −5.34678059630548252631151241078, −5.05047860972900760102176266937, −3.52145953847747792837542252293, −2.65059789339849248574895007838, −0.74990501539926170424215150519,
1.53582975117414575228325259414, 2.70903007137061022312457110735, 4.30611015457652755242854243332, 4.94294645566657088247892055707, 5.95948248938382279961685873443, 6.72175419967052905892492108448, 7.35098662636669391305676982972, 8.207513504444556240677824278251, 9.604523427248471688876601478508, 10.47157001190389780198782601263
