| L(s) = 1 | + (2.02e4 + 3.49e4i)2-s + (2.39e6 − 4.14e6i)3-s + (−5.47e8 + 9.49e8i)4-s + (5.22e7 + 9.05e7i)5-s + 1.93e11·6-s + (1.78e12 + 1.64e11i)7-s + (−2.25e13 + 0.00390i)8-s + (−1.14e13 − 1.98e13i)9-s + (−2.11e12 + 3.65e12i)10-s + (1.12e15 − 1.95e15i)11-s + (2.62e15 + 4.53e15i)12-s − 1.02e16·13-s + (3.03e16 + 6.58e16i)14-s + (5.00e14 + 0.0312i)15-s + (−1.62e17 − 2.80e17i)16-s + (−5.32e17 + 9.22e17i)17-s + ⋯ |
| L(s) = 1 | + (0.871 + 1.51i)2-s + (0.288 − 0.499i)3-s + (−1.02 + 1.76i)4-s + (0.00383 + 0.00663i)5-s + 1.00·6-s + (0.995 + 0.0916i)7-s − 1.81·8-s + (−0.166 − 0.288i)9-s + (−0.00668 + 0.0115i)10-s + (0.896 − 1.55i)11-s + (0.589 + 1.02i)12-s − 0.721·13-s + (0.729 + 1.58i)14-s + 0.00442·15-s + (−0.562 − 0.974i)16-s + (−0.767 + 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.778 + 0.627i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(15)\) |
\(\approx\) |
\(1.954930847\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.954930847\) |
| \(L(\frac{31}{2})\) |
|
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 | \( 1 + (-1.78e12 - 1.64e11i)T \) |
| good | 2 | \( 1 + (-2.02e4 - 3.49e4i)T + (-2.68e8 + 4.64e8i)T^{2} \) |
| 5 | \( 1 + (-5.22e7 - 9.05e7i)T + (-9.31e19 + 1.61e20i)T^{2} \) |
| 11 | \( 1 + (-1.12e15 + 1.95e15i)T + (-7.93e29 - 1.37e30i)T^{2} \) |
| 13 | \( 1 + 1.02e16T + 2.01e32T^{2} \) |
| 17 | \( 1 + (5.32e17 - 9.22e17i)T + (-2.40e35 - 4.17e35i)T^{2} \) |
| 19 | \( 1 + (1.83e18 + 3.17e18i)T + (-6.06e36 + 1.05e37i)T^{2} \) |
| 23 | \( 1 + (4.07e19 + 7.06e19i)T + (-1.54e39 + 2.67e39i)T^{2} \) |
| 29 | \( 1 + 2.92e21T + 2.56e42T^{2} \) |
| 31 | \( 1 + (-5.48e20 + 9.50e20i)T + (-8.88e42 - 1.53e43i)T^{2} \) |
| 37 | \( 1 + (6.68e21 + 1.15e22i)T + (-1.50e45 + 2.60e45i)T^{2} \) |
| 41 | \( 1 + 2.55e23T + 5.89e46T^{2} \) |
| 43 | \( 1 + 5.26e23T + 2.34e47T^{2} \) |
| 47 | \( 1 + (6.24e23 + 1.08e24i)T + (-1.54e48 + 2.68e48i)T^{2} \) |
| 53 | \( 1 + (4.88e24 - 8.46e24i)T + (-5.04e49 - 8.74e49i)T^{2} \) |
| 59 | \( 1 + (-2.77e25 + 4.80e25i)T + (-1.13e51 - 1.95e51i)T^{2} \) |
| 61 | \( 1 + (4.72e25 + 8.19e25i)T + (-2.97e51 + 5.15e51i)T^{2} \) |
| 67 | \( 1 + (2.85e25 - 4.95e25i)T + (-4.52e52 - 7.82e52i)T^{2} \) |
| 71 | \( 1 - 1.01e27T + 4.85e53T^{2} \) |
| 73 | \( 1 + (2.22e26 - 3.84e26i)T + (-5.43e53 - 9.41e53i)T^{2} \) |
| 79 | \( 1 + (8.02e24 + 1.39e25i)T + (-5.37e54 + 9.30e54i)T^{2} \) |
| 83 | \( 1 + 2.53e27T + 4.50e55T^{2} \) |
| 89 | \( 1 + (-1.09e28 - 1.89e28i)T + (-1.70e56 + 2.95e56i)T^{2} \) |
| 97 | \( 1 - 8.82e27T + 4.13e57T^{2} \) |
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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
−12.51534583267368005220104102418, −11.12756572496663703804339430050, −8.666847994394583514618711815067, −8.207065279988437954114772047688, −6.79246656971802748821230631655, −6.02012204375950586116860276392, −4.71722592274333166615039519591, −3.66893469953901929713392955318, −2.00459758449669629063198969723, −0.25564357265389689613843862693,
1.57652294864488761369123737925, 2.07418891369614834049013985425, 3.54879679129309393257532668587, 4.50494577939769517337405940249, 5.20632429548588876225508326846, 7.35789097358579770206416474626, 9.266629052355440869730286435323, 10.04122869983401070541236290625, 11.36208351375865027315667080654, 12.02011785886308891284296555001
