| L(s) = 1 | + (0.618 − 1.90i)2-s + (2.54 − 1.84i)3-s + (−3.23 − 2.35i)4-s + (0.190 + 0.587i)5-s + (−1.94 − 5.98i)6-s + (7.92 + 5.75i)7-s + (−6.47 + 4.70i)8-s + (−5.28 + 16.2i)9-s + 1.23·10-s + (−13.5 + 33.8i)11-s − 12.5·12-s + (4.30 − 13.2i)13-s + (15.8 − 11.5i)14-s + (1.57 + 1.14i)15-s + (4.94 + 15.2i)16-s + (−37.2 − 114. i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (0.489 − 0.355i)3-s + (−0.404 − 0.293i)4-s + (0.0170 + 0.0525i)5-s + (−0.132 − 0.407i)6-s + (0.428 + 0.310i)7-s + (−0.286 + 0.207i)8-s + (−0.195 + 0.602i)9-s + 0.0390·10-s + (−0.372 + 0.927i)11-s − 0.302·12-s + (0.0917 − 0.282i)13-s + (0.302 − 0.219i)14-s + (0.0270 + 0.0196i)15-s + (0.0772 + 0.237i)16-s + (−0.532 − 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.569 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.17153 - 0.613241i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17153 - 0.613241i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.618 + 1.90i)T \) |
| 11 | \( 1 + (13.5 - 33.8i)T \) |
| good | 3 | \( 1 + (-2.54 + 1.84i)T + (8.34 - 25.6i)T^{2} \) |
| 5 | \( 1 + (-0.190 - 0.587i)T + (-101. + 73.4i)T^{2} \) |
| 7 | \( 1 + (-7.92 - 5.75i)T + (105. + 326. i)T^{2} \) |
| 13 | \( 1 + (-4.30 + 13.2i)T + (-1.77e3 - 1.29e3i)T^{2} \) |
| 17 | \( 1 + (37.2 + 114. i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-37.6 + 27.3i)T + (2.11e3 - 6.52e3i)T^{2} \) |
| 23 | \( 1 + 82.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + (166. + 120. i)T + (7.53e3 + 2.31e4i)T^{2} \) |
| 31 | \( 1 + (89.7 - 276. i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-38.9 - 28.3i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-136. + 99.1i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 366.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-179. + 130. i)T + (3.20e4 - 9.87e4i)T^{2} \) |
| 53 | \( 1 + (173. - 534. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (332. + 241. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-93.9 - 289. i)T + (-1.83e5 + 1.33e5i)T^{2} \) |
| 67 | \( 1 - 107.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-21.4 - 66.0i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (66.7 + 48.5i)T + (1.20e5 + 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-391. + 1.20e3i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-256. - 790. i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 - 1.48e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (-119. + 368. i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−17.80027677707644716030348057812, −15.91025571150448948605225966579, −14.45624926645086696580028530724, −13.45552519469780260899113102461, −12.15028133597364417996597594524, −10.76652641205108748274517416475, −9.145017607184501886167251345588, −7.52883520690419941515376047101, −5.00233260091383183310836373755, −2.43964727187547806451304094872,
3.85983701952052033802711773211, 5.98490803082236251686910662354, 7.955421201304318013444905570311, 9.192578294110622090835381382339, 11.02243870997212642398773977482, 12.86356536863772211745159701662, 14.20697489503662999166144055192, 15.07366330043290643621687180134, 16.35305456279572916729969959570, 17.49477045410830729880471869673
