| L(s) = 1 | + (−1.41 + 0.0438i)2-s + (−0.987 + 1.47i)3-s + (1.99 − 0.123i)4-s + (−3.28 + 0.653i)5-s + (1.33 − 2.13i)6-s + (−0.628 + 3.15i)7-s + (−2.81 + 0.262i)8-s + (−0.0604 − 0.145i)9-s + (4.61 − 1.06i)10-s + (1.92 − 1.28i)11-s + (−1.78 + 3.07i)12-s + (−0.0837 + 0.0837i)13-s + (0.749 − 4.49i)14-s + (2.27 − 5.49i)15-s + (3.96 − 0.494i)16-s + (3.85 − 1.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0309i)2-s + (−0.570 + 0.853i)3-s + (0.998 − 0.0619i)4-s + (−1.46 + 0.292i)5-s + (0.543 − 0.870i)6-s + (−0.237 + 1.19i)7-s + (−0.995 + 0.0928i)8-s + (−0.0201 − 0.0486i)9-s + (1.45 − 0.337i)10-s + (0.581 − 0.388i)11-s + (−0.516 + 0.886i)12-s + (−0.0232 + 0.0232i)13-s + (0.200 − 1.20i)14-s + (0.587 − 1.41i)15-s + (0.992 − 0.123i)16-s + (0.934 − 0.356i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.177031 + 0.327277i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.177031 + 0.327277i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0438i)T \) |
| 17 | \( 1 + (-3.85 + 1.47i)T \) |
| good | 3 | \( 1 + (0.987 - 1.47i)T + (-1.14 - 2.77i)T^{2} \) |
| 5 | \( 1 + (3.28 - 0.653i)T + (4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (0.628 - 3.15i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-1.92 + 1.28i)T + (4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (0.0837 - 0.0837i)T - 13iT^{2} \) |
| 19 | \( 1 + (2.00 + 0.829i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-3.36 - 5.03i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-1.83 - 9.24i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (0.139 + 0.0931i)T + (11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (1.34 + 0.897i)T + (14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (2.42 + 0.483i)T + (37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-2.87 + 1.19i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (3.01 + 3.01i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.976 + 2.35i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-4.48 - 10.8i)T + (-41.7 + 41.7i)T^{2} \) |
| 61 | \( 1 + (-1.18 + 5.96i)T + (-56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 - 11.7T + 67T^{2} \) |
| 71 | \( 1 + (-7.46 + 11.1i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (0.178 - 0.0355i)T + (67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (6.42 - 4.29i)T + (30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (1.53 - 3.71i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (1.73 + 1.73i)T + 89iT^{2} \) |
| 97 | \( 1 + (-0.851 - 4.28i)T + (-89.6 + 37.1i)T^{2} \) |
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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
−15.55437925163263490207587652954, −14.75883452740093560985114078097, −12.29799512244250544003424903875, −11.54943162292711680116841915103, −10.78157134827442754602214603836, −9.440212334426383941608593797582, −8.381257358738680111826806351771, −7.05995378521761904080561566938, −5.41751105577698169044252854005, −3.39728493140507523907169513799,
0.78056339960360732824264545390, 3.93001069599093805088010370994, 6.51825435070681333259175294553, 7.37750519484204606886484086006, 8.263648065414067334221408049844, 9.941810680396934085254759680898, 11.18337599194049995062244973520, 12.04186735699538187187875511631, 12.80735665206666400058828104209, 14.67779969413207063067454560234
