| L(s) = 1 | − 3-s + 5-s + 3.21i·7-s + 9-s − 2.53i·11-s + 1.54i·13-s − 15-s − 2.67·17-s + (−1.05 − 4.22i)19-s − 3.21i·21-s − 5.98i·23-s + 25-s − 27-s + 0.925i·29-s − 3.27·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.21i·7-s + 0.333·9-s − 0.765i·11-s + 0.429i·13-s − 0.258·15-s − 0.647·17-s + (−0.241 − 0.970i)19-s − 0.700i·21-s − 1.24i·23-s + 0.200·25-s − 0.192·27-s + 0.171i·29-s − 0.587·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4682836856\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4682836856\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + (1.05 + 4.22i)T \) |
| good | 7 | \( 1 - 3.21iT - 7T^{2} \) |
| 11 | \( 1 + 2.53iT - 11T^{2} \) |
| 13 | \( 1 - 1.54iT - 13T^{2} \) |
| 17 | \( 1 + 2.67T + 17T^{2} \) |
| 23 | \( 1 + 5.98iT - 23T^{2} \) |
| 29 | \( 1 - 0.925iT - 29T^{2} \) |
| 31 | \( 1 + 3.27T + 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 - 9.19iT - 41T^{2} \) |
| 43 | \( 1 - 11.9iT - 43T^{2} \) |
| 47 | \( 1 + 4.08iT - 47T^{2} \) |
| 53 | \( 1 - 2.52iT - 53T^{2} \) |
| 59 | \( 1 + 9.59T + 59T^{2} \) |
| 61 | \( 1 + 3.68T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 + 7.94T + 71T^{2} \) |
| 73 | \( 1 - 7.83T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 + 11.0iT - 83T^{2} \) |
| 89 | \( 1 + 2.71iT - 89T^{2} \) |
| 97 | \( 1 - 8.46iT - 97T^{2} \) |
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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
−8.674694075989502783962413822375, −8.145600677182865555837197475429, −6.92203858013585428628116522076, −6.33928729414445002604436606699, −5.89891696864236277456376031210, −4.92880348358668330724794203088, −4.49136823920211399462223980345, −3.07300580976022623454675060572, −2.44473378638758479278684034314, −1.33505322268717074173876646288,
0.13981930741998909380532053128, 1.40509150887942997853386664511, 2.25036390338312590380558439262, 3.74103441314560338264997110787, 4.07914702033211576774197151323, 5.19563552262216826388290667438, 5.69276091726849807539269647394, 6.61816386915950107892106498092, 7.31777054473305587434599419758, 7.67184605142977502079263951975
