| L(s) = 1 | − 3-s + (1.37 − 1.76i)5-s − i·7-s + 9-s + 1.66i·11-s + 1.73·13-s + (−1.37 + 1.76i)15-s − 7.29i·17-s − 3.24i·19-s + i·21-s + 3.99i·23-s + (−1.22 − 4.84i)25-s − 27-s − 1.30i·29-s − 5.53·31-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + (0.614 − 0.789i)5-s − 0.377i·7-s + 0.333·9-s + 0.502i·11-s + 0.479·13-s + (−0.354 + 0.455i)15-s − 1.76i·17-s − 0.744i·19-s + 0.218i·21-s + 0.832i·23-s + (−0.245 − 0.969i)25-s − 0.192·27-s − 0.242i·29-s − 0.993·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.297602607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.297602607\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.37 + 1.76i)T \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - 1.66iT - 11T^{2} \) |
| 13 | \( 1 - 1.73T + 13T^{2} \) |
| 17 | \( 1 + 7.29iT - 17T^{2} \) |
| 19 | \( 1 + 3.24iT - 19T^{2} \) |
| 23 | \( 1 - 3.99iT - 23T^{2} \) |
| 29 | \( 1 + 1.30iT - 29T^{2} \) |
| 31 | \( 1 + 5.53T + 31T^{2} \) |
| 37 | \( 1 + 0.985T + 37T^{2} \) |
| 41 | \( 1 + 3.50T + 41T^{2} \) |
| 43 | \( 1 - 3.68T + 43T^{2} \) |
| 47 | \( 1 + 1.88iT - 47T^{2} \) |
| 53 | \( 1 - 7.72T + 53T^{2} \) |
| 59 | \( 1 + 6.26iT - 59T^{2} \) |
| 61 | \( 1 - 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 2.45T + 71T^{2} \) |
| 73 | \( 1 + 14.3iT - 73T^{2} \) |
| 79 | \( 1 + 7.63T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 8.15T + 89T^{2} \) |
| 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
−8.477460717035041006932166863439, −7.36498388358720785608305499516, −6.99453244368007907143033272926, −5.95900242660666375006272556972, −5.25478384298140185846399941003, −4.72704355596538855602557925161, −3.80451670384169419975985699516, −2.55840868726176158125341207813, −1.47045470006231263164098527360, −0.43671418576752558448456133634,
1.38088016758693602909657520507, 2.29103946134813018923314139987, 3.43523716349970250801813207717, 4.11885032487034624334759007398, 5.44363533970332141279911285905, 5.86755772492374341515669273295, 6.47518554095970164719548476784, 7.20569841909453389379457882551, 8.276391740870507415678232089965, 8.738877931635944760646899921053
