| L(s) = 1 | + (1.81 − 0.0732i)2-s + (−0.278 − 0.960i)3-s + (1.30 − 0.105i)4-s + (−3.76 + 1.42i)5-s + (−0.576 − 1.72i)6-s + (−1.29 + 3.03i)7-s + (−1.24 + 0.150i)8-s + (−0.845 + 0.534i)9-s + (−6.74 + 2.87i)10-s + (−0.0893 + 0.141i)11-s + (−0.465 − 1.22i)12-s + (−2.65 − 2.43i)13-s + (−2.12 + 5.60i)14-s + (2.41 + 3.21i)15-s + (−4.83 + 0.786i)16-s + (−1.24 − 0.531i)17-s + ⋯ |
| L(s) = 1 | + (1.28 − 0.0518i)2-s + (−0.160 − 0.554i)3-s + (0.654 − 0.0528i)4-s + (−1.68 + 0.638i)5-s + (−0.235 − 0.704i)6-s + (−0.488 + 1.14i)7-s + (−0.438 + 0.0533i)8-s + (−0.281 + 0.178i)9-s + (−2.13 + 0.908i)10-s + (−0.0269 + 0.0426i)11-s + (−0.134 − 0.354i)12-s + (−0.736 − 0.676i)13-s + (−0.568 + 1.49i)14-s + (0.624 + 0.831i)15-s + (−1.20 + 0.196i)16-s + (−0.302 − 0.128i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.227458 + 0.590046i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.227458 + 0.590046i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.278 + 0.960i)T \) |
| 13 | \( 1 + (2.65 + 2.43i)T \) |
| good | 2 | \( 1 + (-1.81 + 0.0732i)T + (1.99 - 0.160i)T^{2} \) |
| 5 | \( 1 + (3.76 - 1.42i)T + (3.74 - 3.31i)T^{2} \) |
| 7 | \( 1 + (1.29 - 3.03i)T + (-4.84 - 5.04i)T^{2} \) |
| 11 | \( 1 + (0.0893 - 0.141i)T + (-4.71 - 9.93i)T^{2} \) |
| 17 | \( 1 + (1.24 + 0.531i)T + (11.7 + 12.2i)T^{2} \) |
| 19 | \( 1 + (-5.53 - 3.19i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.39 + 2.40i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0325 + 0.808i)T + (-28.9 + 2.33i)T^{2} \) |
| 31 | \( 1 + (4.06 - 4.59i)T + (-3.73 - 30.7i)T^{2} \) |
| 37 | \( 1 + (-4.10 - 0.838i)T + (34.0 + 14.5i)T^{2} \) |
| 41 | \( 1 + (4.95 - 1.43i)T + (34.6 - 21.9i)T^{2} \) |
| 43 | \( 1 + (-1.64 - 8.06i)T + (-39.5 + 16.8i)T^{2} \) |
| 47 | \( 1 + (0.289 + 0.199i)T + (16.6 + 43.9i)T^{2} \) |
| 53 | \( 1 + (-1.02 - 8.41i)T + (-51.4 + 12.6i)T^{2} \) |
| 59 | \( 1 + (-0.463 + 2.85i)T + (-55.9 - 18.6i)T^{2} \) |
| 61 | \( 1 + (11.7 + 8.82i)T + (16.9 + 58.5i)T^{2} \) |
| 67 | \( 1 + (0.455 - 5.64i)T + (-66.1 - 10.7i)T^{2} \) |
| 71 | \( 1 + (10.7 + 10.3i)T + (2.85 + 70.9i)T^{2} \) |
| 73 | \( 1 + (-1.49 - 2.84i)T + (-41.4 + 60.0i)T^{2} \) |
| 79 | \( 1 + (-2.80 + 4.06i)T + (-28.0 - 73.8i)T^{2} \) |
| 83 | \( 1 + (-3.96 - 16.0i)T + (-73.4 + 38.5i)T^{2} \) |
| 89 | \( 1 + (1.00 - 0.578i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.48 - 3.66i)T + (19.4 - 95.0i)T^{2} \) |
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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
−11.69755233755891474117053560603, −10.85729884908871638402168984086, −9.445144391624895222907275101946, −8.268193772960489261216050853941, −7.47938761187194173456414770258, −6.50858568428679463578932717368, −5.57647350020076195654274124276, −4.57303822865117311442874969699, −3.32416509898651315699919345113, −2.75574634034174816584484658123,
0.24350924442075922396181944292, 3.22804110497349582271114812582, 4.01462701895301885738554139838, 4.51796206881609620884351108415, 5.45562581014995490732002256464, 6.95444463902819550651960498225, 7.51457023258983551077723925363, 8.854284301745028435134609341902, 9.680582398014116339242391600250, 10.96468070972111513353127128745
