| L(s) = 1 | + (0.877 − 0.479i)2-s + (−0.997 + 0.0713i)3-s + (0.540 − 0.841i)4-s + (−2.09 + 0.779i)5-s + (−0.841 + 0.540i)6-s + (−0.448 + 1.20i)7-s + (0.0713 − 0.997i)8-s + (0.989 − 0.142i)9-s + (−1.46 + 1.68i)10-s + (0.00232 + 0.00793i)11-s + (−0.479 + 0.877i)12-s + (5.34 − 1.99i)13-s + (0.182 + 1.27i)14-s + (2.03 − 0.927i)15-s + (−0.415 − 0.909i)16-s + (1.76 + 0.384i)17-s + ⋯ |
| L(s) = 1 | + (0.620 − 0.338i)2-s + (−0.575 + 0.0411i)3-s + (0.270 − 0.420i)4-s + (−0.937 + 0.348i)5-s + (−0.343 + 0.220i)6-s + (−0.169 + 0.454i)7-s + (0.0252 − 0.352i)8-s + (0.329 − 0.0474i)9-s + (−0.463 + 0.534i)10-s + (0.000702 + 0.00239i)11-s + (−0.138 + 0.253i)12-s + (1.48 − 0.553i)13-s + (0.0488 + 0.339i)14-s + (0.525 − 0.239i)15-s + (−0.103 − 0.227i)16-s + (0.429 + 0.0933i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0399i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.60160 + 0.0320149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60160 + 0.0320149i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.877 + 0.479i)T \) |
| 3 | \( 1 + (0.997 - 0.0713i)T \) |
| 5 | \( 1 + (2.09 - 0.779i)T \) |
| 23 | \( 1 + (-1.70 - 4.48i)T \) |
| good | 7 | \( 1 + (0.448 - 1.20i)T + (-5.29 - 4.58i)T^{2} \) |
| 11 | \( 1 + (-0.00232 - 0.00793i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-5.34 + 1.99i)T + (9.82 - 8.51i)T^{2} \) |
| 17 | \( 1 + (-1.76 - 0.384i)T + (15.4 + 7.06i)T^{2} \) |
| 19 | \( 1 + (-5.16 - 3.31i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-3.74 - 5.82i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-1.20 - 1.38i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (1.50 - 2.00i)T + (-10.4 - 35.5i)T^{2} \) |
| 41 | \( 1 + (-0.636 + 4.42i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (0.451 + 6.31i)T + (-42.5 + 6.11i)T^{2} \) |
| 47 | \( 1 + (-0.432 + 0.432i)T - 47iT^{2} \) |
| 53 | \( 1 + (11.7 + 4.39i)T + (40.0 + 34.7i)T^{2} \) |
| 59 | \( 1 + (3.99 + 1.82i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (-1.16 + 1.01i)T + (8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-2.37 - 4.35i)T + (-36.2 + 56.3i)T^{2} \) |
| 71 | \( 1 + (-13.8 - 4.06i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (2.82 + 13.0i)T + (-66.4 + 30.3i)T^{2} \) |
| 79 | \( 1 + (5.54 - 12.1i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (2.53 + 1.89i)T + (23.3 + 79.6i)T^{2} \) |
| 89 | \( 1 + (1.24 - 1.43i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (4.18 - 3.13i)T + (27.3 - 93.0i)T^{2} \) |
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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
−10.78520592049746134608338214688, −9.910985857935897147965536953487, −8.717592288309095869748995534951, −7.75748640528640772390173347715, −6.78360053569769744348279583474, −5.82524509978041446450179557650, −5.08144030877506732565200235613, −3.71858365363458920838975981730, −3.18663059578340241247444280706, −1.22001094514728718631890259336,
0.967708379511488361601519937201, 3.13511982385672032190416040736, 4.15014689860579458177571130805, 4.85650220245254348141700361849, 6.04067820907324226624273906425, 6.81552474263770100565244804701, 7.70359522982038496562562797854, 8.518629478534082050293418851053, 9.586803377255780237479820770892, 10.84646849704851254226090780260
