| L(s) = 1 | + (0.827 + 1.14i)2-s + (−1.80 + 1.20i)3-s + (−0.631 + 1.89i)4-s + (0.177 − 0.892i)5-s + (−2.86 − 1.06i)6-s + (3.16 − 0.628i)7-s + (−2.69 + 0.846i)8-s + (0.646 − 1.56i)9-s + (1.16 − 0.534i)10-s + (2.03 − 3.04i)11-s + (−1.14 − 4.17i)12-s + (0.674 + 0.674i)13-s + (3.33 + 3.10i)14-s + (0.753 + 1.81i)15-s + (−3.20 − 2.39i)16-s + (−0.463 − 4.09i)17-s + ⋯ |
| L(s) = 1 | + (0.584 + 0.811i)2-s + (−1.03 + 0.694i)3-s + (−0.315 + 0.948i)4-s + (0.0793 − 0.398i)5-s + (−1.17 − 0.436i)6-s + (1.19 − 0.237i)7-s + (−0.954 + 0.299i)8-s + (0.215 − 0.520i)9-s + (0.369 − 0.169i)10-s + (0.613 − 0.918i)11-s + (−0.331 − 1.20i)12-s + (0.186 + 0.186i)13-s + (0.891 + 0.830i)14-s + (0.194 + 0.469i)15-s + (−0.800 − 0.598i)16-s + (−0.112 − 0.993i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0196 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0196 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.658166 + 0.671214i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.658166 + 0.671214i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.827 - 1.14i)T \) |
| 17 | \( 1 + (0.463 + 4.09i)T \) |
| good | 3 | \( 1 + (1.80 - 1.20i)T + (1.14 - 2.77i)T^{2} \) |
| 5 | \( 1 + (-0.177 + 0.892i)T + (-4.61 - 1.91i)T^{2} \) |
| 7 | \( 1 + (-3.16 + 0.628i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-2.03 + 3.04i)T + (-4.20 - 10.1i)T^{2} \) |
| 13 | \( 1 + (-0.674 - 0.674i)T + 13iT^{2} \) |
| 19 | \( 1 + (6.92 - 2.87i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-1.74 - 1.16i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (7.94 + 1.57i)T + (26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (1.92 + 2.88i)T + (-11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (-3.94 - 5.90i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (0.0665 + 0.334i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-7.55 - 3.13i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (4.03 - 4.03i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.49 + 3.61i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (-2.57 + 6.20i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (5.23 - 1.04i)T + (56.3 - 23.3i)T^{2} \) |
| 67 | \( 1 - 0.313T + 67T^{2} \) |
| 71 | \( 1 + (5.04 - 3.37i)T + (27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-1.46 + 7.38i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (1.91 - 2.86i)T + (-30.2 - 72.9i)T^{2} \) |
| 83 | \( 1 + (-1.34 - 3.24i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (2.89 - 2.89i)T - 89iT^{2} \) |
| 97 | \( 1 + (3.78 + 0.753i)T + (89.6 + 37.1i)T^{2} \) |
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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
−15.06970329294486800881365576534, −14.20446802715605324908099167008, −12.99645470910319812900089335016, −11.56703707836027390685590626137, −11.04437104811158475795986976794, −9.156457018652959520818350879424, −7.942138208689164207344729595190, −6.26498491627252398257534088164, −5.16132869883250794576722234822, −4.18838170212546195506297793982,
1.84009143378224748210358501251, 4.41765325542428778542173091063, 5.78487293977176164957260449278, 6.94349951192691466952440380751, 8.885452089742296208726656519415, 10.71439483413060861454716328479, 11.17809081722609188435455389295, 12.34674714789318840793837790448, 12.95030317094413675641042315509, 14.62380793627114857616983770898
