| L(s) = 1 | + (−0.233 + 0.645i)2-s + (−0.604 + 0.261i)3-s + (1.17 + 0.979i)4-s + (−1.02 − 0.0738i)5-s + (−0.0271 − 0.451i)6-s + (0.845 + 0.671i)7-s + (−2.09 + 1.21i)8-s + (−1.75 + 1.86i)9-s + (0.287 − 0.644i)10-s + (0.121 + 0.775i)11-s + (−0.966 − 0.285i)12-s + (−2.57 − 0.829i)13-s + (−0.631 + 0.388i)14-s + (0.639 − 0.223i)15-s + (0.249 + 1.37i)16-s + (−0.00127 − 0.00173i)17-s + ⋯ |
| L(s) = 1 | + (−0.165 + 0.456i)2-s + (−0.349 + 0.150i)3-s + (0.586 + 0.489i)4-s + (−0.458 − 0.0330i)5-s + (−0.0110 − 0.184i)6-s + (0.319 + 0.253i)7-s + (−0.740 + 0.431i)8-s + (−0.586 + 0.622i)9-s + (0.0909 − 0.203i)10-s + (0.0367 + 0.233i)11-s + (−0.278 − 0.0825i)12-s + (−0.713 − 0.229i)13-s + (−0.168 + 0.103i)14-s + (0.165 − 0.0576i)15-s + (0.0623 + 0.342i)16-s + (−0.000309 − 0.000419i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 787 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0668945 - 0.650128i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0668945 - 0.650128i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 787 | \( 1 + (-21.0 + 18.5i)T \) |
| good | 2 | \( 1 + (0.233 - 0.645i)T + (-1.53 - 1.28i)T^{2} \) |
| 3 | \( 1 + (0.604 - 0.261i)T + (2.05 - 2.18i)T^{2} \) |
| 5 | \( 1 + (1.02 + 0.0738i)T + (4.94 + 0.716i)T^{2} \) |
| 7 | \( 1 + (-0.845 - 0.671i)T + (1.58 + 6.81i)T^{2} \) |
| 11 | \( 1 + (-0.121 - 0.775i)T + (-10.4 + 3.37i)T^{2} \) |
| 13 | \( 1 + (2.57 + 0.829i)T + (10.5 + 7.59i)T^{2} \) |
| 17 | \( 1 + (0.00127 + 0.00173i)T + (-5.02 + 16.2i)T^{2} \) |
| 19 | \( 1 + (-2.23 + 2.15i)T + (0.683 - 18.9i)T^{2} \) |
| 23 | \( 1 + (4.00 + 1.50i)T + (17.2 + 15.1i)T^{2} \) |
| 29 | \( 1 + (2.49 - 6.86i)T + (-22.2 - 18.5i)T^{2} \) |
| 31 | \( 1 + (1.14 + 0.165i)T + (29.7 + 8.79i)T^{2} \) |
| 37 | \( 1 + (0.688 - 4.38i)T + (-35.2 - 11.3i)T^{2} \) |
| 41 | \( 1 + (-11.3 - 2.77i)T + (36.3 + 18.9i)T^{2} \) |
| 43 | \( 1 + (-3.00 + 3.04i)T + (-0.515 - 42.9i)T^{2} \) |
| 47 | \( 1 + (5.67 - 4.29i)T + (12.7 - 45.2i)T^{2} \) |
| 53 | \( 1 + (13.7 - 2.67i)T + (49.1 - 19.8i)T^{2} \) |
| 59 | \( 1 + (4.00 + 1.96i)T + (36.1 + 46.6i)T^{2} \) |
| 61 | \( 1 + (2.19 + 5.63i)T + (-44.9 + 41.2i)T^{2} \) |
| 67 | \( 1 + (-11.5 - 8.71i)T + (18.2 + 64.4i)T^{2} \) |
| 71 | \( 1 + (11.3 + 1.09i)T + (69.6 + 13.5i)T^{2} \) |
| 73 | \( 1 + (-2.47 + 0.0594i)T + (72.9 - 3.49i)T^{2} \) |
| 79 | \( 1 + (-4.33 - 3.98i)T + (6.62 + 78.7i)T^{2} \) |
| 83 | \( 1 + (9.43 + 3.55i)T + (62.4 + 54.6i)T^{2} \) |
| 89 | \( 1 + (1.56 + 0.421i)T + (76.8 + 44.8i)T^{2} \) |
| 97 | \( 1 + (1.00 - 2.56i)T + (-71.4 - 65.6i)T^{2} \) |
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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
−11.01442117037879574458741266775, −9.871213614272521232273478345488, −8.840815082417014641418294368802, −7.921599633543360383741028179394, −7.53552020712075936990126394528, −6.41910020659837476122073513824, −5.51354718548958901653408426329, −4.60673057981498447480943502721, −3.22105328076360722159156927148, −2.16302424194317110809502325027,
0.33215004846391872883355688525, 1.82252569890957382319644930408, 3.10816257836087515120310683387, 4.20134789066464076718147760038, 5.63866617251738863022909286387, 6.16022831550494799097282513543, 7.30637901707446186316366031207, 7.974966984030520543741695151964, 9.332431770914614694397802353293, 9.805139784803452506206968084809
