| L(s) = 1 | + (1.65e4 − 2.87e4i)2-s + (−2.39e6 − 4.14e6i)3-s + (−2.81e8 − 4.87e8i)4-s + (1.20e10 − 2.08e10i)5-s − 1.58e11·6-s + (1.59e12 + 8.28e11i)7-s + (−8.68e11 − 0.00292i)8-s + (−1.14e13 + 1.98e13i)9-s + (−3.99e14 − 6.92e14i)10-s + (7.46e14 + 1.29e15i)11-s + (−1.34e15 + 2.33e15i)12-s + 1.20e16·13-s + (5.01e16 − 3.19e16i)14-s + (−1.15e17 − 8i)15-s + (1.36e17 − 2.36e17i)16-s + (−4.62e16 − 8.00e16i)17-s + ⋯ |
| L(s) = 1 | + (0.715 − 1.23i)2-s + (−0.288 − 0.499i)3-s + (−0.524 − 0.908i)4-s + (0.882 − 1.52i)5-s − 0.826·6-s + (0.887 + 0.461i)7-s − 0.0698·8-s + (−0.166 + 0.288i)9-s + (−1.26 − 2.18i)10-s + (0.592 + 1.02i)11-s + (−0.302 + 0.524i)12-s + 0.846·13-s + (1.20 − 0.769i)14-s − 1.01·15-s + (0.474 − 0.821i)16-s + (−0.0666 − 0.115i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.106i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(4.783526335\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.783526335\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (-1.59e12 - 8.28e11i)T \) |
| good | 2 | \( 1 + (-1.65e4 + 2.87e4i)T + (-2.68e8 - 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-1.20e10 + 2.08e10i)T + (-9.31e19 - 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-7.46e14 - 1.29e15i)T + (-7.93e29 + 1.37e30i)T^{2} \) |
| 13 | \( 1 - 1.20e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (4.62e16 + 8.00e16i)T + (-2.40e35 + 4.17e35i)T^{2} \) |
| 19 | \( 1 + (2.51e18 - 4.34e18i)T + (-6.06e36 - 1.05e37i)T^{2} \) |
| 23 | \( 1 + (-3.02e19 + 5.24e19i)T + (-1.54e39 - 2.67e39i)T^{2} \) |
| 29 | \( 1 - 1.71e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (2.84e21 + 4.92e21i)T + (-8.88e42 + 1.53e43i)T^{2} \) |
| 37 | \( 1 + (1.28e22 - 2.22e22i)T + (-1.50e45 - 2.60e45i)T^{2} \) |
| 41 | \( 1 - 4.38e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 2.80e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (-8.39e23 + 1.45e24i)T + (-1.54e48 - 2.68e48i)T^{2} \) |
| 53 | \( 1 + (9.01e23 + 1.56e24i)T + (-5.04e49 + 8.74e49i)T^{2} \) |
| 59 | \( 1 + (4.24e25 + 7.34e25i)T + (-1.13e51 + 1.95e51i)T^{2} \) |
| 61 | \( 1 + (2.58e25 - 4.47e25i)T + (-2.97e51 - 5.15e51i)T^{2} \) |
| 67 | \( 1 + (1.31e26 + 2.28e26i)T + (-4.52e52 + 7.82e52i)T^{2} \) |
| 71 | \( 1 - 4.57e25T + 4.85e53T^{2} \) |
| 73 | \( 1 + (7.88e26 + 1.36e27i)T + (-5.43e53 + 9.41e53i)T^{2} \) |
| 79 | \( 1 + (1.69e27 - 2.93e27i)T + (-5.37e54 - 9.30e54i)T^{2} \) |
| 83 | \( 1 - 1.05e28T + 4.50e55T^{2} \) |
| 89 | \( 1 + (4.73e27 - 8.20e27i)T + (-1.70e56 - 2.95e56i)T^{2} \) |
| 97 | \( 1 - 1.80e28T + 4.13e57T^{2} \) |
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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
−12.03184469797669926863460687413, −10.67256516643716101525598736300, −9.313495776533751985510677188805, −8.127434994801475848262456334654, −6.05992606224217920828008459807, −4.95185006626599317106675918203, −4.21014267872102427107081486676, −2.19490327974001396516750320804, −1.62941851000996047002719402741, −0.863021760897741190484606390876,
1.33131982019482983314481789883, 3.06548361421334627495375471364, 4.25449948449165327926504423327, 5.58320795594127693137288721084, 6.37453949694371991428348208794, 7.25503661883110171561808103142, 8.842185522227054876224312413230, 10.68408218673159818004746762433, 11.08842916269601285078261898157, 13.49724145978547549086789908999
