L(s) = 1 | + (−0.342 − 0.939i)2-s + (0.939 − 0.342i)3-s + (−0.766 + 0.642i)4-s + (−0.893 + 1.06i)5-s + (−0.642 − 0.766i)6-s + (2.63 + 0.266i)7-s + (0.866 + 0.500i)8-s + (0.766 − 0.642i)9-s + (1.30 + 0.475i)10-s + (−1.06 − 1.84i)11-s + (−0.500 + 0.866i)12-s + (−2.48 + 2.08i)13-s + (−0.649 − 2.56i)14-s + (−0.475 + 1.30i)15-s + (0.173 − 0.984i)16-s + (4.52 − 5.39i)17-s + ⋯ |
L(s) = 1 | + (−0.241 − 0.664i)2-s + (0.542 − 0.197i)3-s + (−0.383 + 0.321i)4-s + (−0.399 + 0.475i)5-s + (−0.262 − 0.312i)6-s + (0.994 + 0.100i)7-s + (0.306 + 0.176i)8-s + (0.255 − 0.214i)9-s + (0.412 + 0.150i)10-s + (−0.321 − 0.557i)11-s + (−0.144 + 0.249i)12-s + (−0.690 + 0.579i)13-s + (−0.173 − 0.685i)14-s + (−0.122 + 0.337i)15-s + (0.0434 − 0.246i)16-s + (1.09 − 1.30i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.745 + 0.666i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.55726 - 0.595037i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.55726 - 0.595037i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.342 + 0.939i)T \) |
| 3 | \( 1 + (-0.939 + 0.342i)T \) |
| 7 | \( 1 + (-2.63 - 0.266i)T \) |
| 19 | \( 1 + (-4.04 - 1.63i)T \) |
good | 5 | \( 1 + (0.893 - 1.06i)T + (-0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (1.06 + 1.84i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.48 - 2.08i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.52 + 5.39i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.558 - 3.16i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-8.74 + 1.54i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 - 0.00653T + 31T^{2} \) |
| 37 | \( 1 + (-0.273 + 0.158i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.36 - 4.50i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-10.5 + 3.84i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (8.07 + 9.62i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (2.41 + 2.87i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (0.645 + 0.541i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (5.53 - 0.975i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (2.46 - 6.77i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (4.14 + 11.3i)T + (-54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.67 - 12.8i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (10.1 + 1.78i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (2.99 - 1.72i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (8.46 + 3.07i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-2.09 + 11.8i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.09032390214673370045067153089, −9.431410864670703348770641518832, −8.461313522489456153656716689945, −7.64807354350197464395685558565, −7.18800163010664690294502764636, −5.53882718513976230949887602706, −4.61195462454580753969548703591, −3.35706169317076682014558511008, −2.58797876961687895470275413582, −1.16881224703155933917902869492,
1.18751909317138349267970967324, 2.80722363067115510220011321289, 4.33273375286060322063697103905, 4.86859594559469766936963974915, 5.91942432028924119155816167033, 7.28574839604931110697241891677, 7.951963268197200896174068198761, 8.353414867281310176990479845188, 9.398812626596988452142316760636, 10.23763788786606041072333092122