| L(s) = 1 | + 0.617i·5-s − 2.19i·7-s − 0.928·11-s − 0.951·13-s + 0.493i·17-s + 2.39i·19-s − 5.17·23-s + 4.61·25-s + 4.22i·29-s + 2.47i·31-s + 1.35·35-s − 37-s − 7.06i·41-s − 9.01i·43-s + 0.0135·47-s + ⋯ |
| L(s) = 1 | + 0.276i·5-s − 0.827i·7-s − 0.279·11-s − 0.263·13-s + 0.119i·17-s + 0.548i·19-s − 1.08·23-s + 0.923·25-s + 0.784i·29-s + 0.445i·31-s + 0.228·35-s − 0.164·37-s − 1.10i·41-s − 1.37i·43-s + 0.00197·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1866136550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1866136550\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 5 | \( 1 - 0.617iT - 5T^{2} \) |
| 7 | \( 1 + 2.19iT - 7T^{2} \) |
| 11 | \( 1 + 0.928T + 11T^{2} \) |
| 13 | \( 1 + 0.951T + 13T^{2} \) |
| 17 | \( 1 - 0.493iT - 17T^{2} \) |
| 19 | \( 1 - 2.39iT - 19T^{2} \) |
| 23 | \( 1 + 5.17T + 23T^{2} \) |
| 29 | \( 1 - 4.22iT - 29T^{2} \) |
| 31 | \( 1 - 2.47iT - 31T^{2} \) |
| 41 | \( 1 + 7.06iT - 41T^{2} \) |
| 43 | \( 1 + 9.01iT - 43T^{2} \) |
| 47 | \( 1 - 0.0135T + 47T^{2} \) |
| 53 | \( 1 + 7.85iT - 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 5.10T + 61T^{2} \) |
| 67 | \( 1 + 11.1iT - 67T^{2} \) |
| 71 | \( 1 - 0.138T + 71T^{2} \) |
| 73 | \( 1 + 6.72T + 73T^{2} \) |
| 79 | \( 1 - 10.7iT - 79T^{2} \) |
| 83 | \( 1 + 14.6T + 83T^{2} \) |
| 89 | \( 1 - 13.8iT - 89T^{2} \) |
| 97 | \( 1 - 9.48T + 97T^{2} \) |
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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
−7.76699594280825367456594135529, −7.12193392480035075666845769855, −6.56052118640161279874314851181, −5.63422162346878334496396044834, −4.93363422340362384673676644354, −4.01039125498890510749773415930, −3.41246155839226727885540104175, −2.38013944484151103739343161421, −1.36755026808044177377832756123, −0.04861802548886308689530199366,
1.36788431922562076050787536672, 2.48162844755529464400717007472, 3.04003493796815193640426585336, 4.32761498811511294815933767007, 4.78704692325783074604772427109, 5.79866906349242987884088450878, 6.16739063719165317379788288452, 7.19947986576283974955981693646, 7.86540647954659414008646738888, 8.547072787784256172627306076418
