| L(s) = 1 | + (0.0762 + 0.234i)2-s + (0.809 − 0.587i)3-s + (1.56 − 1.13i)4-s + (−2.09 + 0.784i)5-s + (0.199 + 0.145i)6-s − 1.24·7-s + (0.786 + 0.571i)8-s + (0.309 − 0.951i)9-s + (−0.343 − 0.431i)10-s + (0.794 + 2.44i)11-s + (0.599 − 1.84i)12-s + (−1.44 + 4.45i)13-s + (−0.0950 − 0.292i)14-s + (−1.23 + 1.86i)15-s + (1.12 − 3.46i)16-s + (−4.72 − 3.43i)17-s + ⋯ |
| L(s) = 1 | + (0.0539 + 0.165i)2-s + (0.467 − 0.339i)3-s + (0.784 − 0.569i)4-s + (−0.936 + 0.350i)5-s + (0.0814 + 0.0592i)6-s − 0.471·7-s + (0.278 + 0.201i)8-s + (0.103 − 0.317i)9-s + (−0.108 − 0.136i)10-s + (0.239 + 0.736i)11-s + (0.172 − 0.532i)12-s + (−0.401 + 1.23i)13-s + (−0.0254 − 0.0781i)14-s + (−0.318 + 0.481i)15-s + (0.281 − 0.865i)16-s + (−1.14 − 0.832i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.06122 - 0.0864442i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.06122 - 0.0864442i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 5 | \( 1 + (2.09 - 0.784i)T \) |
| good | 2 | \( 1 + (-0.0762 - 0.234i)T + (-1.61 + 1.17i)T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 11 | \( 1 + (-0.794 - 2.44i)T + (-8.89 + 6.46i)T^{2} \) |
| 13 | \( 1 + (1.44 - 4.45i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (4.72 + 3.43i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (3.37 + 2.45i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.496 - 1.52i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.60 + 1.89i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-7.43 - 5.40i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.394 - 1.21i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-2.68 + 8.27i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 3.88T + 43T^{2} \) |
| 47 | \( 1 + (-2.59 + 1.88i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-10.7 + 7.83i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (1.97 - 6.07i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.18 - 3.63i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-8.19 - 5.95i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (11.2 - 8.15i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.51 + 10.8i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (9.29 - 6.74i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.29 - 1.66i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (0.426 + 1.31i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (0.0246 - 0.0179i)T + (29.9 - 92.2i)T^{2} \) |
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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
−14.69313782167822946717650218122, −13.64823620571171457062894151589, −12.13924877125038950396218507327, −11.41638596186005135259975682935, −10.08635811224722119917497659876, −8.757123612382304020940032699428, −7.05978538827850358696143434189, −6.76609314070461626487614599375, −4.47385452084453862617183599379, −2.48103035249482416270888426644,
2.96370912596098787260026551243, 4.20490053127493811592745988592, 6.36942009081778893560097119743, 7.88142932934350262853324792442, 8.588830684508688743961371119658, 10.33241096898994974216272627405, 11.28588493425255199284640198442, 12.47515159120655407758643030809, 13.19767786682024242391570918921, 14.93865574809768083031200837398
