| L(s) = 1 | + (1.45 − 0.946i)3-s + 5-s − 1.74i·7-s + (1.20 − 2.74i)9-s − 0.702·11-s − 4.70i·13-s + (1.45 − 0.946i)15-s − 1.49i·17-s − 1.07·19-s + (−1.64 − 2.52i)21-s − 2.17i·23-s + 25-s + (−0.849 − 5.12i)27-s + 2.17i·29-s − 3.66i·31-s + ⋯ |
| L(s) = 1 | + (0.837 − 0.546i)3-s + 0.447·5-s − 0.657i·7-s + (0.402 − 0.915i)9-s − 0.211·11-s − 1.30i·13-s + (0.374 − 0.244i)15-s − 0.362i·17-s − 0.247·19-s + (−0.359 − 0.550i)21-s − 0.452i·23-s + 0.200·25-s + (−0.163 − 0.986i)27-s + 0.404i·29-s − 0.658i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.356795997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.356795997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.45 + 0.946i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-7.89 + 2.15i)T \) |
| good | 7 | \( 1 + 1.74iT - 7T^{2} \) |
| 11 | \( 1 + 0.702T + 11T^{2} \) |
| 13 | \( 1 + 4.70iT - 13T^{2} \) |
| 17 | \( 1 + 1.49iT - 17T^{2} \) |
| 19 | \( 1 + 1.07T + 19T^{2} \) |
| 23 | \( 1 + 2.17iT - 23T^{2} \) |
| 29 | \( 1 - 2.17iT - 29T^{2} \) |
| 31 | \( 1 + 3.66iT - 31T^{2} \) |
| 37 | \( 1 + 7.97T + 37T^{2} \) |
| 41 | \( 1 - 1.68T + 41T^{2} \) |
| 43 | \( 1 - 2.43iT - 43T^{2} \) |
| 47 | \( 1 - 8.94iT - 47T^{2} \) |
| 53 | \( 1 + 12.5T + 53T^{2} \) |
| 59 | \( 1 - 9.27iT - 59T^{2} \) |
| 61 | \( 1 + 2.37iT - 61T^{2} \) |
| 71 | \( 1 + 15.2iT - 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 + 2.10iT - 79T^{2} \) |
| 83 | \( 1 - 12.4iT - 83T^{2} \) |
| 89 | \( 1 + 9.88iT - 89T^{2} \) |
| 97 | \( 1 + 10.0iT - 97T^{2} \) |
| 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
−7.954180840956870401194039670238, −7.67429933916454008551234547707, −6.78366235968078368377313467651, −6.12219514390662617658996858395, −5.20262388141014491355623465989, −4.27290720498055939182888181955, −3.29781891133103631045492460322, −2.68202684259621404056923606113, −1.61775219695726768781362700965, −0.55893508724814548569498999061,
1.70902393111981168452467769969, 2.27267844757824410975471657339, 3.29353683003016943658771921342, 4.08054126134898211597519865996, 4.94112826859678194757287049758, 5.61037642366255370065938896951, 6.59848824873495651231129252500, 7.24814006710423305524016088548, 8.282639944833342790085128378224, 8.718338491792499946772855828301
