L(s) = 1 | + (1.20 + 0.692i)2-s − 2.82·3-s + (−0.0395 − 0.0685i)4-s + (−0.449 + 0.259i)5-s + (−3.39 − 1.95i)6-s − 2.88i·8-s + 4.98·9-s − 0.719·10-s + 1.62i·11-s + (0.111 + 0.193i)12-s + (1.42 + 3.31i)13-s + (1.26 − 0.733i)15-s + (1.91 − 3.32i)16-s + (0.974 + 1.68i)17-s + (5.98 + 3.45i)18-s − 2.49i·19-s + ⋯ |
L(s) = 1 | + (0.848 + 0.490i)2-s − 1.63·3-s + (−0.0197 − 0.0342i)4-s + (−0.200 + 0.116i)5-s + (−1.38 − 0.799i)6-s − 1.01i·8-s + 1.66·9-s − 0.227·10-s + 0.489i·11-s + (0.0322 + 0.0559i)12-s + (0.395 + 0.918i)13-s + (0.327 − 0.189i)15-s + (0.479 − 0.830i)16-s + (0.236 + 0.409i)17-s + (1.41 + 0.814i)18-s − 0.571i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00293 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00293 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.726511 + 0.724383i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.726511 + 0.724383i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (-1.42 - 3.31i)T \) |
good | 2 | \( 1 + (-1.20 - 0.692i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + (0.449 - 0.259i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 1.62iT - 11T^{2} \) |
| 17 | \( 1 + (-0.974 - 1.68i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 2.49iT - 19T^{2} \) |
| 23 | \( 1 + (4.57 - 7.91i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.61 - 4.52i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-5.01 - 2.89i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.85 - 5.11i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.64 + 2.10i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.498 - 0.863i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.91 - 2.25i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.44 + 7.70i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.37 + 3.10i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 + 8.37iT - 67T^{2} \) |
| 71 | \( 1 + (4.50 + 2.59i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-10.2 - 5.91i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.491 + 0.850i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 8.91iT - 83T^{2} \) |
| 89 | \( 1 + (-10.4 - 6.00i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.82 - 2.21i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.03230800766720002260673603643, −10.05003129136990146382084995412, −9.369380038031725622863415125222, −7.72620979980010487873569224601, −6.74156024468371339002444070909, −6.23316781378192333533240019861, −5.33457120236282336859674077397, −4.63449158215549846735315608507, −3.70985219947659647812143429800, −1.32305892894279036268625861411,
0.59391530321083812751228959264, 2.63222759022414081679180807034, 4.09119167664243849600221110589, 4.67882859375666792408783875994, 5.92711960672116433772801679825, 6.07674715455267260780251138314, 7.72122054914214451164847989501, 8.441195101672886999036151217459, 10.01155788260598622075154974059, 10.66286479149929580965967937513