| L(s) = 1 | + (−2.08 + 1.51i)2-s + (−1.27 − 3.91i)3-s + (−0.418 + 1.28i)4-s + (0.733 + 0.532i)5-s + (8.59 + 6.24i)6-s + (−2.16 + 6.65i)7-s + (−7.45 − 22.9i)8-s + (8.10 − 5.89i)9-s − 2.33·10-s + (11.5 − 34.6i)11-s + 5.58·12-s + (58.7 − 42.6i)13-s + (−5.57 − 17.1i)14-s + (1.15 − 3.55i)15-s + (41.5 + 30.1i)16-s + (15.4 + 11.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.737 + 0.535i)2-s + (−0.245 − 0.754i)3-s + (−0.0523 + 0.161i)4-s + (0.0655 + 0.0476i)5-s + (0.584 + 0.424i)6-s + (−0.116 + 0.359i)7-s + (−0.329 − 1.01i)8-s + (0.300 − 0.218i)9-s − 0.0738·10-s + (0.315 − 0.948i)11-s + 0.134·12-s + (1.25 − 0.909i)13-s + (−0.106 − 0.327i)14-s + (0.0198 − 0.0611i)15-s + (0.648 + 0.471i)16-s + (0.220 + 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.834835 - 0.285345i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.834835 - 0.285345i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.16 - 6.65i)T \) |
| 11 | \( 1 + (-11.5 + 34.6i)T \) |
| good | 2 | \( 1 + (2.08 - 1.51i)T + (2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (1.27 + 3.91i)T + (-21.8 + 15.8i)T^{2} \) |
| 5 | \( 1 + (-0.733 - 0.532i)T + (38.6 + 118. i)T^{2} \) |
| 13 | \( 1 + (-58.7 + 42.6i)T + (678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (-15.4 - 11.2i)T + (1.51e3 + 4.67e3i)T^{2} \) |
| 19 | \( 1 + (5.67 + 17.4i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 + (-47.0 + 144. i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (216. - 157. i)T + (9.20e3 - 2.83e4i)T^{2} \) |
| 37 | \( 1 + (128. - 396. i)T + (-4.09e4 - 2.97e4i)T^{2} \) |
| 41 | \( 1 + (88.6 + 272. i)T + (-5.57e4 + 4.05e4i)T^{2} \) |
| 43 | \( 1 - 164.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-30.0 - 92.5i)T + (-8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-529. + 385. i)T + (4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (99.1 - 305. i)T + (-1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-32.6 - 23.7i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 - 548.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (305. + 221. i)T + (1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-259. + 800. i)T + (-3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (453. - 329. i)T + (1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (600. + 436. i)T + (1.76e5 + 5.43e5i)T^{2} \) |
| 89 | \( 1 - 872.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (1.19e3 - 867. i)T + (2.82e5 - 8.68e5i)T^{2} \) |
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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
−13.61781218982744474102198297153, −12.84121384527704236651792875358, −11.81368741181891161505111073063, −10.39402502012937403517639329260, −8.945212400420478439429319557114, −8.152702992591434000657541811853, −6.86226929743711644369455491976, −5.93786693772660483156100903184, −3.49787750836433435515184994800, −0.833294873449799550007025344450,
1.58258845352678540965477875810, 4.02365058934726467834648248741, 5.43916016403487045385426762686, 7.19599735782057888060512189834, 8.957477296442273527681558707348, 9.643874601497241751771617286023, 10.71641792823702717992562772569, 11.35650192449791631846876405904, 12.91143967471013964958907809947, 14.20241923945083743481721379955
