L(s) = 1 | + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−1.36 − 1.36i)7-s + (0.707 + 0.707i)8-s − 2.31i·11-s + (3.46 − 3.46i)13-s + 1.93·14-s − 1.00·16-s + (−3.86 + 3.86i)17-s + 0.535i·19-s + (1.63 + 1.63i)22-s + (1.41 + 1.41i)23-s + 4.89i·26-s + (−1.36 + 1.36i)28-s − 3.48·29-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.516 − 0.516i)7-s + (0.250 + 0.250i)8-s − 0.696i·11-s + (0.960 − 0.960i)13-s + 0.516·14-s − 0.250·16-s + (−0.937 + 0.937i)17-s + 0.122i·19-s + (0.348 + 0.348i)22-s + (0.294 + 0.294i)23-s + 0.960i·26-s + (−0.258 + 0.258i)28-s − 0.647·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4847159974\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4847159974\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (1.36 + 1.36i)T + 7iT^{2} \) |
| 11 | \( 1 + 2.31iT - 11T^{2} \) |
| 13 | \( 1 + (-3.46 + 3.46i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.86 - 3.86i)T - 17iT^{2} \) |
| 19 | \( 1 - 0.535iT - 19T^{2} \) |
| 23 | \( 1 + (-1.41 - 1.41i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.48T + 29T^{2} \) |
| 31 | \( 1 + 10.6T + 31T^{2} \) |
| 37 | \( 1 + (8.19 + 8.19i)T + 37iT^{2} \) |
| 41 | \( 1 - 9.52iT - 41T^{2} \) |
| 43 | \( 1 + (-4.73 + 4.73i)T - 43iT^{2} \) |
| 47 | \( 1 + (-4.24 + 4.24i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.26 - 2.26i)T + 53iT^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 2.53T + 61T^{2} \) |
| 67 | \( 1 + (4.19 + 4.19i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.14iT - 71T^{2} \) |
| 73 | \( 1 + (7.63 - 7.63i)T - 73iT^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 + (11.2 + 11.2i)T + 83iT^{2} \) |
| 89 | \( 1 + 8.76T + 89T^{2} \) |
| 97 | \( 1 + (-1.43 - 1.43i)T + 97iT^{2} \) |
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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
−9.035061413702334575759470656989, −8.682332649206212060104127741997, −7.67522853356599778602362970253, −6.98525427005216395844223692445, −5.99431731261003277831513222844, −5.52290431553860559555344422936, −4.05194006652804130658693817636, −3.29851389502508644577260425038, −1.68404400516956108229716674765, −0.23148371473036718811291667978,
1.61859404661630340489084635154, 2.60850162206498348692226594100, 3.72146338419279997867683997781, 4.64736755295292341175996740359, 5.79748202580182870564651108227, 6.83809159589770252936210614707, 7.34434295970367078185007006482, 8.677818186509973392326077801803, 9.085418150191034236963954325402, 9.678036387833605543398158773839