| L(s) = 1 | − 2-s + (−1.57 − 0.721i)3-s + 4-s + (1.57 + 0.721i)6-s + (−2.29 + 1.31i)7-s − 8-s + (1.95 + 2.27i)9-s + 3.91i·11-s + (−1.57 − 0.721i)12-s + 4.99·13-s + (2.29 − 1.31i)14-s + 16-s − 3.54i·17-s + (−1.95 − 2.27i)18-s − 3.14i·19-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + (−0.909 − 0.416i)3-s + 0.5·4-s + (0.642 + 0.294i)6-s + (−0.867 + 0.496i)7-s − 0.353·8-s + (0.652 + 0.757i)9-s + 1.18i·11-s + (−0.454 − 0.208i)12-s + 1.38·13-s + (0.613 − 0.351i)14-s + 0.250·16-s − 0.860i·17-s + (−0.461 − 0.535i)18-s − 0.722i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1390993803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1390993803\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (1.57 + 0.721i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.29 - 1.31i)T \) |
| good | 11 | \( 1 - 3.91iT - 11T^{2} \) |
| 13 | \( 1 - 4.99T + 13T^{2} \) |
| 17 | \( 1 + 3.54iT - 17T^{2} \) |
| 19 | \( 1 + 3.14iT - 19T^{2} \) |
| 23 | \( 1 + 7.54T + 23T^{2} \) |
| 29 | \( 1 + 7.54iT - 29T^{2} \) |
| 31 | \( 1 - 4.19iT - 31T^{2} \) |
| 37 | \( 1 - 10.4iT - 37T^{2} \) |
| 41 | \( 1 + 9.32T + 41T^{2} \) |
| 43 | \( 1 + 2.91iT - 43T^{2} \) |
| 47 | \( 1 - 8.00iT - 47T^{2} \) |
| 53 | \( 1 - 0.288T + 53T^{2} \) |
| 59 | \( 1 + 5.89T + 59T^{2} \) |
| 61 | \( 1 - 2.48iT - 61T^{2} \) |
| 67 | \( 1 + 0.545iT - 67T^{2} \) |
| 71 | \( 1 - 5.37iT - 71T^{2} \) |
| 73 | \( 1 + 4.85T + 73T^{2} \) |
| 79 | \( 1 + 0.742T + 79T^{2} \) |
| 83 | \( 1 - 4.45iT - 83T^{2} \) |
| 89 | \( 1 + 12.3T + 89T^{2} \) |
| 97 | \( 1 - 0.524T + 97T^{2} \) |
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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
−10.07508843605080476982975310478, −9.703167029473257804057454775382, −8.619791787906157245633358262628, −7.75601274531972350027928633955, −6.74980195702679290290685709425, −6.34755391207800609318181981888, −5.36454614884632464507226842689, −4.19870136121843766643557589938, −2.71330614966276294001226399420, −1.49724285723152647848134786468,
0.094531384661864704082119839797, 1.45157294166072298863451466773, 3.48279266548690812115790120659, 3.92335314544215167195198370250, 5.68952072968939164442990364438, 6.08165103000007559974774871679, 6.85816062669440935952278628294, 8.042047325252749743342934003376, 8.799016508443171110715652946609, 9.680546747899923892762334046528
