L(s) = 1 | + 0.0439·2-s + (0.820 + 1.42i)3-s − 1.99·4-s + (0.864 − 1.49i)5-s + (0.0360 + 0.0624i)6-s + (−2.17 − 3.76i)7-s − 0.175·8-s + (0.155 − 0.268i)9-s + (0.0380 − 0.0658i)10-s + (0.377 − 0.654i)11-s + (−1.63 − 2.83i)12-s + (0.5 − 0.866i)13-s + (−0.0956 − 0.165i)14-s + 2.83·15-s + 3.98·16-s + (−1.81 − 3.14i)17-s + ⋯ |
L(s) = 1 | + 0.0310·2-s + (0.473 + 0.820i)3-s − 0.999·4-s + (0.386 − 0.670i)5-s + (0.0147 + 0.0254i)6-s + (−0.822 − 1.42i)7-s − 0.0621·8-s + (0.0516 − 0.0895i)9-s + (0.0120 − 0.0208i)10-s + (0.113 − 0.197i)11-s + (−0.472 − 0.819i)12-s + (0.138 − 0.240i)13-s + (−0.0255 − 0.0442i)14-s + 0.732·15-s + 0.997·16-s + (−0.440 − 0.763i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.927570 - 0.625967i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.927570 - 0.625967i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-4.77 - 2.86i)T \) |
good | 2 | \( 1 - 0.0439T + 2T^{2} \) |
| 3 | \( 1 + (-0.820 - 1.42i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.864 + 1.49i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (2.17 + 3.76i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.377 + 0.654i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.81 + 3.14i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.12 + 1.95i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 5.51T + 23T^{2} \) |
| 29 | \( 1 - 2.34T + 29T^{2} \) |
| 37 | \( 1 + (-0.0716 - 0.124i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.10 + 10.5i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.21 + 3.83i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 3.15T + 47T^{2} \) |
| 53 | \( 1 + (4.33 - 7.51i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.89 - 6.74i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + (1.21 - 2.10i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.49 + 4.31i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.56 + 11.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.89 + 6.74i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.47 - 12.9i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 1.15T + 89T^{2} \) |
| 97 | \( 1 - 16.4T + 97T^{2} \) |
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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
−10.60377781329175664465664059304, −10.06367953378224345943546801356, −9.258960066826014251284463616882, −8.758700609707371327197596732749, −7.48319000735830738077404558735, −6.25141324410663070187060767337, −4.85479568134882951765606832565, −4.15997034695528062666963573250, −3.27955223824671369268079148711, −0.72228314067277208989267595042,
2.00085627105561788313153569468, 3.06249057387930870786473735478, 4.53311357090816324746094548713, 6.02076486543305868662132452139, 6.47952605802605349927249746114, 8.012527531691868683508565682847, 8.552788690190633756177130911509, 9.611820174762486268233813465814, 10.20008729708289873455821439555, 11.65520612986086712624089083409