L(s) = 1 | + (−3.04 − 4.76i)2-s + (−6.36 + 6.36i)3-s + (−13.4 + 29.0i)4-s + (25.2 + 25.2i)5-s + (49.7 + 10.9i)6-s − 124. i·7-s + (179. − 24.3i)8-s − 81i·9-s + (43.5 − 197. i)10-s + (−241. − 241. i)11-s + (−99.1 − 270. i)12-s + (215. − 215. i)13-s + (−593. + 379. i)14-s − 321.·15-s + (−662. − 781. i)16-s − 1.34e3·17-s + ⋯ |
L(s) = 1 | + (−0.538 − 0.842i)2-s + (−0.408 + 0.408i)3-s + (−0.420 + 0.907i)4-s + (0.452 + 0.452i)5-s + (0.563 + 0.124i)6-s − 0.960i·7-s + (0.990 − 0.134i)8-s − 0.333i·9-s + (0.137 − 0.624i)10-s + (−0.600 − 0.600i)11-s + (−0.198 − 0.542i)12-s + (0.353 − 0.353i)13-s + (−0.809 + 0.517i)14-s − 0.369·15-s + (−0.646 − 0.762i)16-s − 1.12·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.443338 - 0.726435i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.443338 - 0.726435i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (3.04 + 4.76i)T \) |
| 3 | \( 1 + (6.36 - 6.36i)T \) |
good | 5 | \( 1 + (-25.2 - 25.2i)T + 3.12e3iT^{2} \) |
| 7 | \( 1 + 124. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (241. + 241. i)T + 1.61e5iT^{2} \) |
| 13 | \( 1 + (-215. + 215. i)T - 3.71e5iT^{2} \) |
| 17 | \( 1 + 1.34e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-1.85e3 + 1.85e3i)T - 2.47e6iT^{2} \) |
| 23 | \( 1 + 2.91e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-1.12e3 + 1.12e3i)T - 2.05e7iT^{2} \) |
| 31 | \( 1 - 4.21e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-3.90e3 - 3.90e3i)T + 6.93e7iT^{2} \) |
| 41 | \( 1 - 1.42e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (1.12e4 + 1.12e4i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + 1.30e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (9.96e3 + 9.96e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + (5.31e3 + 5.31e3i)T + 7.14e8iT^{2} \) |
| 61 | \( 1 + (1.72e4 - 1.72e4i)T - 8.44e8iT^{2} \) |
| 67 | \( 1 + (-3.78e4 + 3.78e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 - 2.60e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 2.10e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 9.53e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-6.60e3 + 6.60e3i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 - 1.32e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 7.39e4T + 8.58e9T^{2} \) |
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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
−13.88273585697913691045982125058, −13.14379461645415841973670693588, −11.47265100351022453658017171318, −10.67653157736308708096071211688, −9.836959512879220870961905130252, −8.336713591286851432431432112971, −6.71436182051258440847610971562, −4.61813729833095365538998511506, −2.90600699167472917888002505509, −0.57568514755893846409116640694,
1.66677508781926504777641526113, 5.08143174957016234890660757690, 6.06511308557390810045166195032, 7.53308666987786419478320169348, 8.857983286420991266942241008056, 9.911193210394541381578750544489, 11.49287745492879342428600459957, 12.88691110190613665129550889311, 13.92721383528881083948282484286, 15.36317344228687236262957127772