L(s) = 1 | + (1.11 − 1.92i)2-s + (−0.337 + 0.584i)3-s + (−1.47 − 2.55i)4-s + 1.70·5-s + (0.751 + 1.30i)6-s + (2.45 + 4.24i)7-s − 2.12·8-s + (1.27 + 2.20i)9-s + (1.89 − 3.27i)10-s + (1.43 − 2.48i)11-s + 1.99·12-s + (−3.48 + 0.929i)13-s + 10.9·14-s + (−0.574 + 0.994i)15-s + (0.589 − 1.02i)16-s + (−1.82 − 3.16i)17-s + ⋯ |
L(s) = 1 | + (0.786 − 1.36i)2-s + (−0.194 + 0.337i)3-s + (−0.738 − 1.27i)4-s + 0.760·5-s + (0.306 + 0.531i)6-s + (0.927 + 1.60i)7-s − 0.751·8-s + (0.424 + 0.734i)9-s + (0.598 − 1.03i)10-s + (0.432 − 0.749i)11-s + 0.575·12-s + (−0.966 + 0.257i)13-s + 2.91·14-s + (−0.148 + 0.256i)15-s + (0.147 − 0.255i)16-s + (−0.443 − 0.767i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 + 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.493 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.97785 - 1.15186i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.97785 - 1.15186i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (3.48 - 0.929i)T \) |
| 31 | \( 1 + T \) |
good | 2 | \( 1 + (-1.11 + 1.92i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.337 - 0.584i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 + (-2.45 - 4.24i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.43 + 2.48i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.82 + 3.16i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.68 + 2.91i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.08 - 1.88i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.61 - 7.99i)T + (-14.5 - 25.1i)T^{2} \) |
| 37 | \( 1 + (-4.66 + 8.08i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.86 + 6.68i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.18 + 10.7i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 6.20T + 53T^{2} \) |
| 59 | \( 1 + (4.08 + 7.06i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.07 - 5.32i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.24 - 3.89i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.99 - 3.45i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 6.02T + 73T^{2} \) |
| 79 | \( 1 - 2.87T + 79T^{2} \) |
| 83 | \( 1 + 2.83T + 83T^{2} \) |
| 89 | \( 1 + (4.83 - 8.37i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.81 - 6.60i)T + (-48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.23255896024084775316308088606, −10.57898020397771598035777971134, −9.422284758092643405996131520757, −8.903973382859162403617706092080, −7.34092247366987972969531603445, −5.46945388547776127859468714679, −5.34001433869746491591703211331, −4.13389017426074057140137169394, −2.47714884113523268947804575017, −1.95516666497524004404112349615,
1.61989435054506108005350335980, 4.11732207300432411122278321201, 4.51552903909702675544966487975, 5.92042833757461433061067980898, 6.61587781778556682760829702129, 7.48392324957353326206225888308, 8.036905231424080261239251293225, 9.673197155712161336396445525046, 10.33156689696691496012592931261, 11.62158311375946106811544270776