L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.965 − 0.258i)3-s + (−0.866 + 0.499i)4-s + i·6-s + (0.258 − 2.63i)7-s + (0.707 + 0.707i)8-s + (0.866 + 0.499i)9-s + (1.17 + 2.04i)11-s + (0.965 − 0.258i)12-s + (0.0968 − 0.0968i)13-s + (−2.61 + 0.431i)14-s + (0.500 − 0.866i)16-s + (0.833 − 3.11i)17-s + (0.258 − 0.965i)18-s + (0.434 − 0.752i)19-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.557 − 0.149i)3-s + (−0.433 + 0.249i)4-s + 0.408i·6-s + (0.0978 − 0.995i)7-s + (0.249 + 0.249i)8-s + (0.288 + 0.166i)9-s + (0.355 + 0.615i)11-s + (0.278 − 0.0747i)12-s + (0.0268 − 0.0268i)13-s + (−0.697 + 0.115i)14-s + (0.125 − 0.216i)16-s + (0.202 − 0.754i)17-s + (0.0610 − 0.227i)18-s + (0.0996 − 0.172i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.770 + 0.637i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9621715749\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9621715749\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.258 + 2.63i)T \) |
good | 11 | \( 1 + (-1.17 - 2.04i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.0968 + 0.0968i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.833 + 3.11i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-0.434 + 0.752i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.11 + 0.833i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + 4.08iT - 29T^{2} \) |
| 31 | \( 1 + (0.747 - 0.431i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.54 + 9.51i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 6.86iT - 41T^{2} \) |
| 43 | \( 1 + (-2.57 - 2.57i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.833 + 0.223i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.16 + 8.07i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (5.35 + 9.27i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (7.44 + 4.29i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (12.2 + 3.28i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 - 5.13T + 71T^{2} \) |
| 73 | \( 1 + (15.7 + 4.20i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.02 - 2.32i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.13 + 7.13i)T - 83iT^{2} \) |
| 89 | \( 1 + (-4.98 + 8.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.4 + 11.4i)T + 97iT^{2} \) |
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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
−9.743607822275510740201210484266, −9.034608069853338934757194882711, −7.76971122542629473098738026073, −7.22767871816508794850023976656, −6.25529260502613927617225424951, −4.98814196903291833945961266554, −4.32278611835982891181756353279, −3.21331535925000857094979800848, −1.79282370505397805040633406141, −0.54575216813679515097828070339,
1.39344580644312816297141941292, 3.08805661345799418978857488317, 4.31471037264885991574249690674, 5.38985575638105807982257255531, 5.90833206527891466377255403317, 6.74440879160986860757020418466, 7.72576728434009462201850745240, 8.747119150200903365698062123353, 9.100951957090937710820617920070, 10.24401057892307012154124298928