L(s) = 1 | + (−0.309 − 0.951i)2-s + (2.42 + 1.76i)3-s + (0.809 − 0.587i)4-s + (0.309 − 0.951i)5-s + (0.927 − 2.85i)6-s + (2.42 − 1.76i)7-s + (−2.42 − 1.76i)8-s + (1.85 + 5.70i)9-s − 0.999·10-s + 3.00·12-s + (1.23 + 3.80i)13-s + (−2.42 − 1.76i)14-s + (2.42 − 1.76i)15-s + (−0.309 + 0.951i)16-s + (4.85 − 3.52i)18-s + (−3.23 − 2.35i)19-s + ⋯ |
L(s) = 1 | + (−0.218 − 0.672i)2-s + (1.40 + 1.01i)3-s + (0.404 − 0.293i)4-s + (0.138 − 0.425i)5-s + (0.378 − 1.16i)6-s + (0.917 − 0.666i)7-s + (−0.858 − 0.623i)8-s + (0.618 + 1.90i)9-s − 0.316·10-s + 0.866·12-s + (0.342 + 1.05i)13-s + (−0.648 − 0.471i)14-s + (0.626 − 0.455i)15-s + (−0.0772 + 0.237i)16-s + (1.14 − 0.831i)18-s + (−0.742 − 0.539i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.37909 - 0.710951i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.37909 - 0.710951i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.309 + 0.951i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.309 + 0.951i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-2.42 - 1.76i)T + (0.927 + 2.85i)T^{2} \) |
| 7 | \( 1 + (-2.42 + 1.76i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-1.23 - 3.80i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (3.23 + 2.35i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + (4.85 - 3.52i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.618 + 1.90i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-6.47 + 4.70i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.04 - 2.93i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (-2.42 - 1.76i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.23 - 3.80i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-1.61 + 1.17i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (3.39 - 10.4i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + 13T + 67T^{2} \) |
| 71 | \( 1 + (-0.618 + 1.90i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-6.47 + 4.70i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.09 - 9.51i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-1.23 + 3.80i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - T + 89T^{2} \) |
| 97 | \( 1 + (2.47 + 7.60i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.63105723178609384179110491042, −9.543153920717872048752370919641, −9.173241459075668913966186087691, −8.217353090568468414843307820781, −7.33949572748974404850200086492, −5.91131485224818000266982901215, −4.40238840880682856611799400786, −3.97231325903907202300397747043, −2.52442023018752714886391843180, −1.64543658180506032343725931973,
1.91095190029464593981894167460, 2.60996089031176532691576586150, 3.76351380540971861461186752261, 5.72032501444492930582601360601, 6.41085001414077972736575446037, 7.56909427982818580786916189681, 8.005555007689480603994003117512, 8.476363989806113589840588787306, 9.460552662053755506265634742558, 10.73672643209328905350385699294