| L(s) = 1 | + (0.402 − 1.35i)2-s + (0.480 + 3.19i)3-s + (−1.67 − 1.09i)4-s + (3.18 + 1.24i)5-s + (4.51 + 0.631i)6-s + (2.64 + 0.174i)7-s + (−2.15 + 1.83i)8-s + (−7.07 + 2.18i)9-s + (2.97 − 3.81i)10-s + (0.905 − 2.93i)11-s + (2.67 − 5.87i)12-s + (−4.15 + 0.947i)13-s + (1.29 − 3.50i)14-s + (−2.45 + 10.7i)15-s + (1.62 + 3.65i)16-s + (0.121 − 0.0829i)17-s + ⋯ |
| L(s) = 1 | + (0.284 − 0.958i)2-s + (0.277 + 1.84i)3-s + (−0.838 − 0.545i)4-s + (1.42 + 0.558i)5-s + (1.84 + 0.257i)6-s + (0.997 + 0.0657i)7-s + (−0.761 + 0.648i)8-s + (−2.35 + 0.727i)9-s + (0.940 − 1.20i)10-s + (0.273 − 0.885i)11-s + (0.771 − 1.69i)12-s + (−1.15 + 0.262i)13-s + (0.346 − 0.937i)14-s + (−0.633 + 2.77i)15-s + (0.405 + 0.914i)16-s + (0.0295 − 0.0201i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 - 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.86252 + 0.593309i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.86252 + 0.593309i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.402 + 1.35i)T \) |
| 7 | \( 1 + (-2.64 - 0.174i)T \) |
| good | 3 | \( 1 + (-0.480 - 3.19i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-3.18 - 1.24i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.905 + 2.93i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (4.15 - 0.947i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.121 + 0.0829i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-1.42 + 0.822i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.949 + 0.647i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (1.72 - 3.57i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-3.54 + 6.14i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.09 + 0.456i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (1.33 + 1.67i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (0.298 + 0.237i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (2.35 - 2.18i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (11.4 + 0.856i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-13.6 + 5.35i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (6.74 - 0.505i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (13.1 + 7.59i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.09 + 1.97i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (1.25 + 1.16i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-1.61 - 2.79i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.80 - 1.55i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-10.9 + 3.37i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 9.85T + 97T^{2} \) |
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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
−11.05762974592628346044770032229, −10.53715387653674936545962266132, −9.591651252244837697969359868555, −9.344749675927896309029238729738, −8.201876168555891492427065256334, −6.02746556183706217092536958721, −5.21555595089585453534622226206, −4.45839421119215563708551480071, −3.16109963640075008631632142206, −2.21128776897266912060712035314,
1.38091333638967962532176598908, 2.50743552895557740379051596355, 4.80395765844175257538223393295, 5.64781219854679860882443068708, 6.55599901876467327898959815460, 7.43909728611863309588818850758, 8.088613632520846856752654934929, 9.052493769025063557860941346697, 9.930542784652601790923437363673, 11.82567935980705644266102258535
