| L(s) = 1 | + (1.77e4 − 3.07e4i)2-s + (2.39e6 + 4.14e6i)3-s + (−3.63e8 − 6.29e8i)4-s + (6.21e9 − 1.07e10i)5-s + (1.70e11 − 1.52e−5i)6-s + (−3.33e10 − 1.79e12i)7-s + (−6.74e12 + 0.00585i)8-s + (−1.14e13 + 1.98e13i)9-s + (−2.20e14 − 3.82e14i)10-s + (−9.62e14 − 1.66e15i)11-s + (1.73e15 − 3.00e15i)12-s − 1.69e14·13-s + (−5.58e16 − 3.08e16i)14-s + (5.94e16 + 4i)15-s + (7.51e16 − 1.30e17i)16-s + (−6.79e16 − 1.17e17i)17-s + ⋯ |
| L(s) = 1 | + (0.767 − 1.32i)2-s + (0.288 + 0.499i)3-s + (−0.676 − 1.17i)4-s + (0.455 − 0.788i)5-s + 0.885·6-s + (−0.0186 − 0.999i)7-s − 0.542·8-s + (−0.166 + 0.288i)9-s + (−0.698 − 1.20i)10-s + (−0.764 − 1.32i)11-s + (0.390 − 0.676i)12-s − 0.0119·13-s + (−1.34 − 0.742i)14-s + 0.525·15-s + (0.260 − 0.451i)16-s + (−0.0978 − 0.169i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(2.842944333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.842944333\) |
| \(L(\frac{31}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.39e6 - 4.14e6i)T \) |
| 7 | \( 1 + (3.33e10 + 1.79e12i)T \) |
| good | 2 | \( 1 + (-1.77e4 + 3.07e4i)T + (-2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-6.21e9 + 1.07e10i)T + (-9.31e19 - 1.61e20i)T^{2} \) |
| 11 | \( 1 + (9.62e14 + 1.66e15i)T + (-7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 + 1.69e14T + 2.01e32T^{2} \) |
| 17 | \( 1 + (6.79e16 + 1.17e17i)T + (-2.40e35 + 4.17e35i)T^{2} \) |
| 19 | \( 1 + (1.07e18 - 1.86e18i)T + (-6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-7.99e17 + 1.38e18i)T + (-1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 - 9.05e20T + 2.56e42T^{2} \) |
| 31 | \( 1 + (2.71e21 + 4.70e21i)T + (-8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (6.35e21 - 1.10e22i)T + (-1.50e45 - 2.60e45i)T^{2} \) |
| 41 | \( 1 - 1.65e23T + 5.89e46T^{2} \) |
| 43 | \( 1 - 5.07e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (7.12e23 - 1.23e24i)T + (-1.54e48 - 2.68e48i)T^{2} \) |
| 53 | \( 1 + (-2.30e24 - 3.99e24i)T + (-5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (3.36e25 + 5.82e25i)T + (-1.13e51 + 1.95e51i)T^{2} \) |
| 61 | \( 1 + (-1.48e24 + 2.57e24i)T + (-2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (1.30e26 + 2.26e26i)T + (-4.52e52 + 7.82e52i)T^{2} \) |
| 71 | \( 1 + 1.18e27T + 4.85e53T^{2} \) |
| 73 | \( 1 + (-7.27e26 - 1.26e27i)T + (-5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (1.08e27 - 1.88e27i)T + (-5.37e54 - 9.30e54i)T^{2} \) |
| 83 | \( 1 - 8.08e26T + 4.50e55T^{2} \) |
| 89 | \( 1 + (6.49e27 - 1.12e28i)T + (-1.70e56 - 2.95e56i)T^{2} \) |
| 97 | \( 1 - 7.21e27T + 4.13e57T^{2} \) |
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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
−11.20903344782403074026850788046, −10.47166152949764989516463407255, −9.345957975300777336314098327724, −7.913871072168412103264898280239, −5.73297236346534363991248774827, −4.65896501443826362876174768149, −3.70595440564888185380378765284, −2.67546399145404360843267749682, −1.35642585221061277986036556297, −0.40552297363646957057733357543,
1.91716395478434691672527660788, 2.88341229023191589449055194313, 4.61282833903084227042016999262, 5.72233720762340703903652100283, 6.70946641816046937137835982773, 7.53738182326386642359213659233, 8.842750618847590801018495063581, 10.43160534066130877734514386324, 12.31651076036624847026113855429, 13.19585648275155466818414710954
