L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.913 − 0.406i)3-s + (0.309 + 0.951i)4-s + (0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (−2.44 + 2.71i)7-s + (0.309 − 0.951i)8-s + (0.669 + 0.743i)9-s + (−0.913 + 0.406i)10-s + (−4.12 − 0.876i)11-s + (0.104 − 0.994i)12-s + (−0.492 − 4.68i)13-s + (3.57 − 0.759i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + (6.16 − 1.30i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.527 − 0.234i)3-s + (0.154 + 0.475i)4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (−0.924 + 1.02i)7-s + (0.109 − 0.336i)8-s + (0.223 + 0.247i)9-s + (−0.288 + 0.128i)10-s + (−1.24 − 0.264i)11-s + (0.0301 − 0.287i)12-s + (−0.136 − 1.29i)13-s + (0.955 − 0.203i)14-s + (−0.208 + 0.151i)15-s + (−0.202 + 0.146i)16-s + (1.49 − 0.317i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.738291 + 0.0654602i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.738291 + 0.0654602i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.913 + 0.406i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-1.82 - 5.26i)T \) |
good | 7 | \( 1 + (2.44 - 2.71i)T + (-0.731 - 6.96i)T^{2} \) |
| 11 | \( 1 + (4.12 + 0.876i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (0.492 + 4.68i)T + (-12.7 + 2.70i)T^{2} \) |
| 17 | \( 1 + (-6.16 + 1.30i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (0.488 - 4.65i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-0.0646 + 0.198i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-7.23 - 5.25i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (0.469 + 0.812i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.10 - 1.82i)T + (27.4 - 30.4i)T^{2} \) |
| 43 | \( 1 + (0.868 - 8.26i)T + (-42.0 - 8.94i)T^{2} \) |
| 47 | \( 1 + (-6.16 + 4.47i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-5.61 - 6.23i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (-3.15 - 1.40i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 - 6.42T + 61T^{2} \) |
| 67 | \( 1 + (-5.06 + 8.77i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.53 - 7.26i)T + (-7.42 + 70.6i)T^{2} \) |
| 73 | \( 1 + (0.618 + 0.131i)T + (66.6 + 29.6i)T^{2} \) |
| 79 | \( 1 + (-2.09 + 0.445i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-7.22 + 3.21i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-4.52 - 13.9i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (0.0901 + 0.277i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.31037367920324742913222075352, −9.416356018533861005560243963813, −8.316723628404744728509525223217, −7.891669672080858473410724089705, −6.64299032147386200728979198854, −5.59401460603679362296625853723, −5.21914075019445566087247760485, −3.31434202199306548894198281187, −2.61329671477653158470206579974, −0.961970457176416656232490523281,
0.58969512752545236508461933107, 2.42927895160789370177565034204, 3.79450462112910842544379416651, 4.88729689375079875268219140793, 5.91201803093779186509752326366, 6.80120293366139972712066049514, 7.27737329630426760973784267575, 8.293498552376215169703494164080, 9.517469163152583853850077094169, 10.07255893571383306847530985956