| L(s) = 1 | + (0.618 − 1.61i)3-s − 5-s + (2.61 + 0.381i)7-s + (−2.23 − 2.00i)9-s + 5.23i·11-s + 3.23i·13-s + (−0.618 + 1.61i)15-s − 4.47·17-s + 2.76i·19-s + (2.23 − 4i)21-s + 5.70i·23-s + 25-s + (−4.61 + 2.38i)27-s + 4i·29-s + 1.23i·31-s + ⋯ |
| L(s) = 1 | + (0.356 − 0.934i)3-s − 0.447·5-s + (0.989 + 0.144i)7-s + (−0.745 − 0.666i)9-s + 1.57i·11-s + 0.897i·13-s + (−0.159 + 0.417i)15-s − 1.08·17-s + 0.634i·19-s + (0.487 − 0.872i)21-s + 1.19i·23-s + 0.200·25-s + (−0.888 + 0.458i)27-s + 0.742i·29-s + 0.222i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.385798813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.385798813\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.618 + 1.61i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (-2.61 - 0.381i)T \) |
| good | 11 | \( 1 - 5.23iT - 11T^{2} \) |
| 13 | \( 1 - 3.23iT - 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 - 2.76iT - 19T^{2} \) |
| 23 | \( 1 - 5.70iT - 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 1.23iT - 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 12.4T + 41T^{2} \) |
| 43 | \( 1 - 2.76T + 43T^{2} \) |
| 47 | \( 1 + 1.23T + 47T^{2} \) |
| 53 | \( 1 - 4.76iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 + 12.9iT - 61T^{2} \) |
| 67 | \( 1 - 3.70T + 67T^{2} \) |
| 71 | \( 1 + 10.1iT - 71T^{2} \) |
| 73 | \( 1 - 11.2iT - 73T^{2} \) |
| 79 | \( 1 + 1.52T + 79T^{2} \) |
| 83 | \( 1 - 2.76T + 83T^{2} \) |
| 89 | \( 1 + 3.52T + 89T^{2} \) |
| 97 | \( 1 + 8.18iT - 97T^{2} \) |
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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
−9.235886652324522338612827614426, −8.614765118517576827171206099700, −7.81935755079856211077691446634, −7.13402793633175358308954255486, −6.63708129276080101280675686715, −5.32351079568217471620385797060, −4.53155128408846868815102788063, −3.53903140052386535940547256334, −2.07364294670019352007350621316, −1.62024205455019223989948020937,
0.48821373610816665523348653012, 2.36262855935930725509617555557, 3.31172664782695011499126349848, 4.20133572310169644039791848700, 4.98032197052496609761655702988, 5.74713529357994039564655809887, 6.86375083563695644328614564339, 8.033646159715655973332715404323, 8.479411761334435357461113815639, 8.923117650997363835510549276588
