| L(s) = 1 | + (−2.18 − 0.465i)2-s + (−0.139 + 1.32i)3-s + (2.74 + 1.22i)4-s + (−0.579 − 0.643i)5-s + (0.922 − 2.84i)6-s + (−0.647 − 2.56i)7-s + (−1.81 − 1.32i)8-s + (1.19 + 0.253i)9-s + (0.969 + 1.67i)10-s + (−2.00 + 3.47i)12-s + (−1.04 − 3.21i)13-s + (0.224 + 5.91i)14-s + (0.935 − 0.679i)15-s + (−0.656 − 0.729i)16-s + (2.14 − 0.456i)17-s + (−2.48 − 1.10i)18-s + ⋯ |
| L(s) = 1 | + (−1.54 − 0.328i)2-s + (−0.0805 + 0.766i)3-s + (1.37 + 0.611i)4-s + (−0.259 − 0.287i)5-s + (0.376 − 1.15i)6-s + (−0.244 − 0.969i)7-s + (−0.642 − 0.467i)8-s + (0.397 + 0.0844i)9-s + (0.306 + 0.530i)10-s + (−0.578 + 1.00i)12-s + (−0.289 − 0.891i)13-s + (0.0599 + 1.58i)14-s + (0.241 − 0.175i)15-s + (−0.164 − 0.182i)16-s + (0.520 − 0.110i)17-s + (−0.586 − 0.261i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 + 0.602i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 + 0.602i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0810028 - 0.241651i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0810028 - 0.241651i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.647 + 2.56i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (2.18 + 0.465i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (0.139 - 1.32i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + (0.579 + 0.643i)T + (-0.522 + 4.97i)T^{2} \) |
| 13 | \( 1 + (1.04 + 3.21i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.14 + 0.456i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (3.27 - 1.45i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (0.834 - 1.44i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (1.23 - 0.900i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (5.93 - 6.59i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (0.322 + 3.06i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (0.532 + 0.386i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 9.76T + 43T^{2} \) |
| 47 | \( 1 + (-12.0 + 5.36i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-4.39 + 4.87i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (13.3 + 5.92i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (-3.87 - 4.30i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (2.64 + 4.57i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.42 + 7.45i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (5.40 + 2.40i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (11.4 + 2.44i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (0.174 - 0.536i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (1.76 - 3.05i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.29 + 13.2i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.00398186503010946680073842818, −9.180790210360930382638693565399, −8.318294184693515358673259806906, −7.55209028087659870841947233743, −6.86737710484639008937535423791, −5.33944861235273651902267189713, −4.27289414479505546191156681712, −3.25949559195368436081539487128, −1.61760565256490444839026110559, −0.21681313521693448474007392073,
1.49606858274808793900937279520, 2.46966289826256156997675325721, 4.16765236686885216695481632824, 5.77428202129349322756309620398, 6.62171647609036326165063316670, 7.25123472814242579705250798312, 7.956035686730327003187962014731, 8.887622971592699515885523681780, 9.445161846681022535821417441979, 10.26947900939479888188813828120
