| L(s) = 1 | + (−3.36 + 2.16i)2-s + 5.19i·3-s + (6.60 − 14.5i)4-s + (−11.2 − 17.4i)6-s + 36.6i·7-s + (9.40 + 63.3i)8-s − 27·9-s + 56.2i·11-s + (75.7 + 34.3i)12-s + 219.·13-s + (−79.5 − 123. i)14-s + (−168. − 192. i)16-s + 54.4·17-s + (90.7 − 58.5i)18-s − 155. i·19-s + ⋯ |
| L(s) = 1 | + (−0.840 + 0.541i)2-s + 0.577i·3-s + (0.412 − 0.910i)4-s + (−0.312 − 0.485i)6-s + 0.748i·7-s + (0.146 + 0.989i)8-s − 0.333·9-s + 0.464i·11-s + (0.525 + 0.238i)12-s + 1.29·13-s + (−0.405 − 0.629i)14-s + (−0.659 − 0.751i)16-s + 0.188·17-s + (0.280 − 0.180i)18-s − 0.431i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.910 - 0.412i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.090067011\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.090067011\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.36 - 2.16i)T \) |
| 3 | \( 1 - 5.19iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 36.6iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 56.2iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 219.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 54.4T + 8.35e4T^{2} \) |
| 19 | \( 1 + 155. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 322. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 989.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 847. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 1.83e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.47e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 171. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 3.34e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.74e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.78e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 5.03e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 4.01e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 5.38e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 6.55e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 1.14e4iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 8.23e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 5.36e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.29e4T + 8.85e7T^{2} \) |
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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
−11.22324014707944449887682378933, −10.41978868606947059211049087729, −9.443953689979097918059310414000, −8.761259642635321047382759286429, −7.898170623055586970973694069727, −6.58539141821042981884808690317, −5.73399759286384395347846215255, −4.63043246881731450768035701917, −2.92334488781484490337546377778, −1.34315407107810487354302912245,
0.49445094344109818377437160798, 1.50947628505250425346345858132, 3.02636310263459600554160253550, 4.14019260981962578492467131626, 6.06369646796122278813576461305, 6.95791652926750644637087680984, 8.063275472434687791751949980831, 8.613260200929956348515278771590, 9.875052393332624521389020420146, 10.72746651369745463432187942989
