| L(s) = 1 | + (−3.44 − 4.31i)2-s + (3.65 − 1.75i)3-s + (−5.00 + 21.9i)4-s + (−12.6 + 6.11i)5-s + (−20.1 − 9.70i)6-s + (−15.1 + 10.6i)7-s + (72.0 − 34.6i)8-s + (−6.59 + 8.26i)9-s + (70.1 + 33.7i)10-s + (−21.5 − 27.0i)11-s + (20.2 + 88.8i)12-s + (11.5 + 14.4i)13-s + (98.0 + 28.8i)14-s + (−35.6 + 44.6i)15-s + (−235. − 113. i)16-s + (−12.0 − 52.8i)17-s + ⋯ |
| L(s) = 1 | + (−1.21 − 1.52i)2-s + (0.702 − 0.338i)3-s + (−0.625 + 2.74i)4-s + (−1.13 + 0.546i)5-s + (−1.37 − 0.660i)6-s + (−0.818 + 0.573i)7-s + (3.18 − 1.53i)8-s + (−0.244 + 0.306i)9-s + (2.21 + 1.06i)10-s + (−0.591 − 0.741i)11-s + (0.487 + 2.13i)12-s + (0.245 + 0.308i)13-s + (1.87 + 0.551i)14-s + (−0.613 + 0.768i)15-s + (−3.68 − 1.77i)16-s + (−0.172 − 0.754i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.116824 + 0.0935734i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.116824 + 0.0935734i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (15.1 - 10.6i)T \) |
| good | 2 | \( 1 + (3.44 + 4.31i)T + (-1.78 + 7.79i)T^{2} \) |
| 3 | \( 1 + (-3.65 + 1.75i)T + (16.8 - 21.1i)T^{2} \) |
| 5 | \( 1 + (12.6 - 6.11i)T + (77.9 - 97.7i)T^{2} \) |
| 11 | \( 1 + (21.5 + 27.0i)T + (-296. + 1.29e3i)T^{2} \) |
| 13 | \( 1 + (-11.5 - 14.4i)T + (-488. + 2.14e3i)T^{2} \) |
| 17 | \( 1 + (12.0 + 52.8i)T + (-4.42e3 + 2.13e3i)T^{2} \) |
| 19 | \( 1 + 61.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + (20.4 - 89.5i)T + (-1.09e4 - 5.27e3i)T^{2} \) |
| 29 | \( 1 + (-1.60 - 7.01i)T + (-2.19e4 + 1.05e4i)T^{2} \) |
| 31 | \( 1 + 58.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + (4.07 + 17.8i)T + (-4.56e4 + 2.19e4i)T^{2} \) |
| 41 | \( 1 + (-282. + 136. i)T + (4.29e4 - 5.38e4i)T^{2} \) |
| 43 | \( 1 + (-51.7 - 24.8i)T + (4.95e4 + 6.21e4i)T^{2} \) |
| 47 | \( 1 + (208. + 261. i)T + (-2.31e4 + 1.01e5i)T^{2} \) |
| 53 | \( 1 + (94.1 - 412. i)T + (-1.34e5 - 6.45e4i)T^{2} \) |
| 59 | \( 1 + (629. + 303. i)T + (1.28e5 + 1.60e5i)T^{2} \) |
| 61 | \( 1 + (-89.6 - 392. i)T + (-2.04e5 + 9.84e4i)T^{2} \) |
| 67 | \( 1 - 359.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-40.7 + 178. i)T + (-3.22e5 - 1.55e5i)T^{2} \) |
| 73 | \( 1 + (357. - 448. i)T + (-8.65e4 - 3.79e5i)T^{2} \) |
| 79 | \( 1 - 654.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (286. - 359. i)T + (-1.27e5 - 5.57e5i)T^{2} \) |
| 89 | \( 1 + (388. - 487. i)T + (-1.56e5 - 6.87e5i)T^{2} \) |
| 97 | \( 1 + 388.T + 9.12e5T^{2} \) |
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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
−15.67298283250756264970908167617, −13.73579456090224470246690592308, −12.70580821782936775787853705965, −11.55170106835145940101064126595, −10.79220632291878478961933250411, −9.286744576069170791686990780983, −8.345471402664486830451233347958, −7.39016525867902903284034624425, −3.56194916169758727023895356419, −2.56492400804199397613121143487,
0.14969318129018350600145584471, 4.34631750705223010443515133703, 6.33966472510789972750561982097, 7.72257488364271521802376768972, 8.480299946321708623788092564291, 9.566675160622143820722467402435, 10.66686199701311707844479072858, 12.86409583207845399610080714581, 14.41889690264277349714479846300, 15.30708779746486047638903806359
