L(s) = 1 | + (−1.72 − 0.188i)3-s + 3.09·5-s + (−1.38 − 0.800i)7-s + (2.92 + 0.650i)9-s + (2.72 − 4.71i)11-s + (−3.69 + 2.13i)13-s + (−5.32 − 0.584i)15-s + (1.10 − 0.638i)17-s + (0.117 + 0.204i)19-s + (2.23 + 1.63i)21-s + (4.09 − 2.36i)23-s + 4.57·25-s + (−4.91 − 1.67i)27-s + (−1.17 − 0.676i)29-s + (−1.21 − 0.701i)31-s + ⋯ |
L(s) = 1 | + (−0.994 − 0.109i)3-s + 1.38·5-s + (−0.523 − 0.302i)7-s + (0.976 + 0.216i)9-s + (0.821 − 1.42i)11-s + (−1.02 + 0.591i)13-s + (−1.37 − 0.150i)15-s + (0.268 − 0.154i)17-s + (0.0270 + 0.0468i)19-s + (0.487 + 0.357i)21-s + (0.853 − 0.492i)23-s + 0.914·25-s + (−0.946 − 0.322i)27-s + (−0.217 − 0.125i)29-s + (−0.218 − 0.126i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.17233 - 0.620036i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.17233 - 0.620036i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 + 0.188i)T \) |
| 67 | \( 1 + (-6.48 - 4.99i)T \) |
good | 5 | \( 1 - 3.09T + 5T^{2} \) |
| 7 | \( 1 + (1.38 + 0.800i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.72 + 4.71i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.69 - 2.13i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.10 + 0.638i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.117 - 0.204i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.09 + 2.36i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.17 + 0.676i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.21 + 0.701i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.33 - 5.76i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.91 + 6.77i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 7.91iT - 43T^{2} \) |
| 47 | \( 1 + (-0.523 - 0.302i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 8.20T + 53T^{2} \) |
| 59 | \( 1 + 13.8iT - 59T^{2} \) |
| 61 | \( 1 + (-7.23 + 4.17i)T + (30.5 - 52.8i)T^{2} \) |
| 71 | \( 1 + (-11.3 - 6.53i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.23 + 5.60i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.34 + 4.81i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (12.5 - 7.22i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 2.12iT - 89T^{2} \) |
| 97 | \( 1 + (-2.75 + 1.58i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.02511211801097740588031735350, −9.574638265853316549523925435856, −8.633708422637480978705058559250, −7.12574831440552811029279890948, −6.54197088140277752893517256479, −5.77985522466037636445296978608, −5.04522454229555454905095970239, −3.72633153663204150935259208243, −2.23903044822926545852618301516, −0.824647521971646981977873415716,
1.40520170477513866851480857293, 2.62511886358322185093742383275, 4.28666157878715780133827051731, 5.25535723013203840833620831270, 5.90795472978017887430104515960, 6.78913976941829034492620354394, 7.47383217795255212316659851003, 9.206225966911120399816188193526, 9.702388520659103018018258090628, 10.12568766982780416345601071414