| L(s) = 1 | + (−19.7 − 60.8i)2-s + (−1.60e3 − 1.16e3i)3-s + (−3.31e3 + 2.40e3i)4-s + (100. − 308. i)5-s + (−3.91e4 + 1.20e5i)6-s + (5.16e4 − 3.75e4i)7-s + (2.12e5 + 1.54e5i)8-s + (7.20e5 + 2.21e6i)9-s − 2.07e4·10-s + (5.07e6 + 2.95e6i)11-s + 8.11e6·12-s + (5.82e6 + 1.79e7i)13-s + (−3.30e6 − 2.40e6i)14-s + (−5.19e5 + 3.77e5i)15-s + (5.18e6 − 1.59e7i)16-s + (1.90e7 − 5.85e7i)17-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−1.26 − 0.922i)3-s + (−0.404 + 0.293i)4-s + (0.00286 − 0.00882i)5-s + (−0.342 + 1.05i)6-s + (0.165 − 0.120i)7-s + (0.286 + 0.207i)8-s + (0.451 + 1.39i)9-s − 0.00656·10-s + (0.864 + 0.503i)11-s + 0.784·12-s + (0.334 + 1.02i)13-s + (−0.117 − 0.0852i)14-s + (−0.0117 + 0.00856i)15-s + (0.0772 − 0.237i)16-s + (0.191 − 0.588i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0551 + 0.998i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.0551 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(0.9990823744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9990823744\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (19.7 + 60.8i)T \) |
| 11 | \( 1 + (-5.07e6 - 2.95e6i)T \) |
| good | 3 | \( 1 + (1.60e3 + 1.16e3i)T + (4.92e5 + 1.51e6i)T^{2} \) |
| 5 | \( 1 + (-100. + 308. i)T + (-9.87e8 - 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-5.16e4 + 3.75e4i)T + (2.99e10 - 9.21e10i)T^{2} \) |
| 13 | \( 1 + (-5.82e6 - 1.79e7i)T + (-2.45e14 + 1.78e14i)T^{2} \) |
| 17 | \( 1 + (-1.90e7 + 5.85e7i)T + (-8.01e15 - 5.82e15i)T^{2} \) |
| 19 | \( 1 + (5.21e7 + 3.79e7i)T + (1.29e16 + 3.99e16i)T^{2} \) |
| 23 | \( 1 + 9.10e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (-4.46e8 + 3.24e8i)T + (3.17e18 - 9.75e18i)T^{2} \) |
| 31 | \( 1 + (1.84e9 + 5.69e9i)T + (-1.97e19 + 1.43e19i)T^{2} \) |
| 37 | \( 1 + (-1.46e10 + 1.06e10i)T + (7.52e19 - 2.31e20i)T^{2} \) |
| 41 | \( 1 + (-1.74e10 - 1.26e10i)T + (2.85e20 + 8.79e20i)T^{2} \) |
| 43 | \( 1 - 2.43e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-5.23e10 - 3.80e10i)T + (1.68e21 + 5.19e21i)T^{2} \) |
| 53 | \( 1 + (1.30e10 + 4.02e10i)T + (-2.10e22 + 1.53e22i)T^{2} \) |
| 59 | \( 1 + (2.66e10 - 1.93e10i)T + (3.24e22 - 9.98e22i)T^{2} \) |
| 61 | \( 1 + (3.17e10 - 9.77e10i)T + (-1.30e23 - 9.51e22i)T^{2} \) |
| 67 | \( 1 - 1.42e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-3.21e11 + 9.88e11i)T + (-9.42e23 - 6.84e23i)T^{2} \) |
| 73 | \( 1 + (3.61e11 - 2.62e11i)T + (5.16e23 - 1.59e24i)T^{2} \) |
| 79 | \( 1 + (1.14e12 + 3.50e12i)T + (-3.77e24 + 2.74e24i)T^{2} \) |
| 83 | \( 1 + (-9.93e11 + 3.05e12i)T + (-7.17e24 - 5.21e24i)T^{2} \) |
| 89 | \( 1 + 1.98e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-3.38e12 - 1.04e13i)T + (-5.44e25 + 3.95e25i)T^{2} \) |
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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
−14.22192871547763324657638374812, −12.84095436421495806738745111044, −11.82490981295643803267511052812, −11.07685266143018202216843322620, −9.376562825397032338785687196980, −7.42839690548305415375897512138, −6.13043515539000542700205296502, −4.36296804738691664020762223860, −1.90022380762048458989006933620, −0.72632038947262541799627159325,
0.77718827804811659375683863413, 3.97126567945220057365738117837, 5.42364824241936399216804358204, 6.38371860714893195290341305874, 8.419309505186471325353669523355, 10.01033994627508694591728435008, 11.00949908134244194021711477478, 12.40801468726927906061788719887, 14.33200521946046155873102948458, 15.57694331300495968790282237061
