| L(s) = 1 | + (−0.850 + 0.526i)2-s + (−0.273 − 0.961i)3-s + (0.445 − 0.895i)4-s + (−1.14 + 0.445i)5-s + (0.739 + 0.673i)6-s + (−0.243 + 2.62i)7-s + (0.0922 + 0.995i)8-s + (−0.850 + 0.526i)9-s + (0.742 − 0.983i)10-s + (−1.96 + 1.21i)11-s + (−0.982 − 0.183i)12-s + (0.417 − 4.51i)13-s + (−1.17 − 2.36i)14-s + (0.742 + 0.983i)15-s + (−0.602 − 0.798i)16-s + (2.34 − 2.13i)17-s + ⋯ |
| L(s) = 1 | + (−0.601 + 0.372i)2-s + (−0.157 − 0.555i)3-s + (0.222 − 0.447i)4-s + (−0.513 + 0.199i)5-s + (0.301 + 0.275i)6-s + (−0.0919 + 0.992i)7-s + (0.0326 + 0.352i)8-s + (−0.283 + 0.175i)9-s + (0.234 − 0.310i)10-s + (−0.593 + 0.367i)11-s + (−0.283 − 0.0530i)12-s + (0.115 − 1.25i)13-s + (−0.314 − 0.630i)14-s + (0.191 + 0.253i)15-s + (−0.150 − 0.199i)16-s + (0.568 − 0.517i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.706 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.106640 - 0.257024i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.106640 - 0.257024i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.850 - 0.526i)T \) |
| 3 | \( 1 + (0.273 + 0.961i)T \) |
| 103 | \( 1 + (7.70 + 6.60i)T \) |
| good | 5 | \( 1 + (1.14 - 0.445i)T + (3.69 - 3.36i)T^{2} \) |
| 7 | \( 1 + (0.243 - 2.62i)T + (-6.88 - 1.28i)T^{2} \) |
| 11 | \( 1 + (1.96 - 1.21i)T + (4.90 - 9.84i)T^{2} \) |
| 13 | \( 1 + (-0.417 + 4.51i)T + (-12.7 - 2.38i)T^{2} \) |
| 17 | \( 1 + (-2.34 + 2.13i)T + (1.56 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.57 + 5.53i)T + (-16.1 - 10.0i)T^{2} \) |
| 23 | \( 1 + (3.04 + 1.88i)T + (10.2 + 20.5i)T^{2} \) |
| 29 | \( 1 + (8.91 - 3.45i)T + (21.4 - 19.5i)T^{2} \) |
| 31 | \( 1 + (4.98 + 6.60i)T + (-8.48 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-3.06 - 0.573i)T + (34.5 + 13.3i)T^{2} \) |
| 41 | \( 1 + (7.12 + 2.75i)T + (30.2 + 27.6i)T^{2} \) |
| 43 | \( 1 + (4.88 - 0.912i)T + (40.0 - 15.5i)T^{2} \) |
| 47 | \( 1 + 12.0T + 47T^{2} \) |
| 53 | \( 1 + (-2.99 + 10.5i)T + (-45.0 - 27.9i)T^{2} \) |
| 59 | \( 1 + (-1.04 - 11.2i)T + (-57.9 + 10.8i)T^{2} \) |
| 61 | \( 1 + (-9.97 + 9.09i)T + (5.62 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-1.13 - 12.2i)T + (-65.8 + 12.3i)T^{2} \) |
| 71 | \( 1 + (-4.77 - 1.84i)T + (52.4 + 47.8i)T^{2} \) |
| 73 | \( 1 + (11.8 + 4.58i)T + (53.9 + 49.1i)T^{2} \) |
| 79 | \( 1 + (-14.8 + 5.76i)T + (58.3 - 53.2i)T^{2} \) |
| 83 | \( 1 + (-0.856 + 9.24i)T + (-81.5 - 15.2i)T^{2} \) |
| 89 | \( 1 + (1.43 + 2.87i)T + (-53.6 + 71.0i)T^{2} \) |
| 97 | \( 1 + (9.79 + 8.92i)T + (8.95 + 96.5i)T^{2} \) |
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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
−10.17630483974417524756506545586, −9.354950602727489576281281931568, −8.356157814566972259587958454785, −7.66516211790060104622167980487, −6.97598525320004393928933814710, −5.70942411348908654362608972608, −5.19972368210391039761817981372, −3.29376002687850285070759190879, −2.13037065331127153943228288287, −0.18482136027989819363722887217,
1.66062630863593982171690836291, 3.58334666601411543828495479886, 4.00684230789183582546720453395, 5.40681117997301750194423187688, 6.61294829544954810078887566573, 7.72935071710706785093190276330, 8.259073920182627352894865930780, 9.466238765277418802216582788630, 10.05807676407656652176782527786, 10.85437044883620479414114853689
