L(s) = 1 | + (−0.309 + 0.951i)2-s + (−0.809 + 0.587i)3-s + (−0.809 − 0.587i)4-s + (−0.309 − 0.951i)6-s + (−1.61 − 1.17i)7-s + (0.809 − 0.587i)8-s + (0.309 − 0.951i)9-s + 0.999·12-s + (−1.23 + 3.80i)13-s + (1.61 − 1.17i)14-s + (0.309 + 0.951i)16-s + (−1.85 − 5.70i)17-s + (0.809 + 0.587i)18-s + (3.23 − 2.35i)19-s + 2·21-s + ⋯ |
L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.467 + 0.339i)3-s + (−0.404 − 0.293i)4-s + (−0.126 − 0.388i)6-s + (−0.611 − 0.444i)7-s + (0.286 − 0.207i)8-s + (0.103 − 0.317i)9-s + 0.288·12-s + (−0.342 + 1.05i)13-s + (0.432 − 0.314i)14-s + (0.0772 + 0.237i)16-s + (−0.449 − 1.38i)17-s + (0.190 + 0.138i)18-s + (0.742 − 0.539i)19-s + 0.436·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 726 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.890948 - 0.0211853i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.890948 - 0.0211853i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 - 0.951i)T \) |
| 3 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + (-4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (1.61 + 1.17i)T + (2.16 + 6.65i)T^{2} \) |
| 13 | \( 1 + (1.23 - 3.80i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (1.85 + 5.70i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-3.23 + 2.35i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + (4.85 + 3.52i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.47 + 7.60i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-8.09 - 5.87i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (4.85 - 3.52i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (-4.85 + 3.52i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.47 - 7.60i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + (-1.85 - 5.70i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.61 + 1.17i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-4.32 + 13.3i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (3.70 + 11.4i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + (-4.32 + 13.3i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−10.13278646627346999731399122973, −9.473499696272167564765247583856, −8.913283473706369842653587144656, −7.47692911025604384204181646230, −6.93804619865300375829426504685, −6.09958842756044598156427870740, −4.93686228117197904030314312749, −4.26903959578231737198238953682, −2.78319205162696449398740399811, −0.63143845551932186441852531751,
1.17194181145582999800350966133, 2.66456478718956119617035485346, 3.64128699491492708113598050355, 5.07248366684773933719838114514, 5.84580967773729256827151250219, 6.95305011347541921482079488049, 7.87139298688488579074446544649, 8.862820505973078316772479666273, 9.595476188422447955237605956720, 10.73145562313146554255644422340