| L(s) = 1 | + (−1.01 − 1.72i)2-s + (1.34 − 2.68i)3-s + (−1.92 + 3.50i)4-s + (4.99 + 0.0579i)5-s + (−5.98 + 0.413i)6-s + (−1.78 + 1.78i)7-s + (7.99 − 0.249i)8-s + (−5.37 − 7.21i)9-s + (−4.98 − 8.66i)10-s + (15.7 + 11.4i)11-s + (6.80 + 9.88i)12-s + (2.86 − 18.0i)13-s + (4.87 + 1.25i)14-s + (6.88 − 13.3i)15-s + (−8.56 − 13.5i)16-s + (−11.7 − 23.1i)17-s + ⋯ |
| L(s) = 1 | + (−0.508 − 0.860i)2-s + (0.448 − 0.893i)3-s + (−0.481 + 0.876i)4-s + (0.999 + 0.0115i)5-s + (−0.997 + 0.0688i)6-s + (−0.254 + 0.254i)7-s + (0.999 − 0.0312i)8-s + (−0.597 − 0.801i)9-s + (−0.498 − 0.866i)10-s + (1.43 + 1.03i)11-s + (0.567 + 0.823i)12-s + (0.220 − 1.39i)13-s + (0.348 + 0.0895i)14-s + (0.458 − 0.888i)15-s + (−0.535 − 0.844i)16-s + (−0.693 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.856554 - 1.42504i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.856554 - 1.42504i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.01 + 1.72i)T \) |
| 3 | \( 1 + (-1.34 + 2.68i)T \) |
| 5 | \( 1 + (-4.99 - 0.0579i)T \) |
| good | 7 | \( 1 + (1.78 - 1.78i)T - 49iT^{2} \) |
| 11 | \( 1 + (-15.7 - 11.4i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (-2.86 + 18.0i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (11.7 + 23.1i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (0.167 - 0.515i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (-29.5 + 4.67i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (-5.62 - 17.3i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (3.32 + 1.07i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (18.7 + 2.97i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (20.9 + 28.8i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (29.1 + 29.1i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-3.20 + 6.28i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (11.8 - 23.2i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (12.6 + 17.4i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (41.4 + 30.1i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (-51.0 - 100. i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (-15.3 - 47.2i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (-77.0 + 12.2i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (24.9 + 76.8i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-33.2 - 65.2i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-57.7 - 41.9i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (-51.6 + 101. i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−11.27944420311231432771971893600, −10.13364373070113975419144461621, −9.196261912949425524329279884175, −8.797199249842293931212117505716, −7.33273742259801211227715420274, −6.61684796065864577216702040922, −5.06357946381387575224936640642, −3.29315697194393023379435603880, −2.26152726290883485626889491018, −1.03590318345537083824107418165,
1.64279123389098043666248701267, 3.70707601387049299131823008452, 4.82459525481190749778315623160, 6.17683099948112290487406311659, 6.67274420358003751085319220163, 8.412017994074128127089641196081, 9.039266014774958910853606116256, 9.581713343917107385352525727947, 10.62801477944829233656026596212, 11.37617988594751230172026537299
