| L(s) = 1 | + (1.63 + 0.504i)3-s + (0.140 + 0.358i)5-s + (−1.74 − 3.02i)7-s + (−0.0539 − 0.0367i)9-s + (3.90 − 1.88i)11-s + (1.26 + 0.190i)13-s + (0.0493 + 0.658i)15-s + (0.205 − 0.523i)17-s + (6.30 − 4.29i)19-s + (−1.33 − 5.83i)21-s + (−0.553 + 7.39i)23-s + (3.55 − 3.29i)25-s + (−3.27 − 4.10i)27-s + (3.26 − 1.00i)29-s + (0.717 + 0.665i)31-s + ⋯ |
| L(s) = 1 | + (0.945 + 0.291i)3-s + (0.0629 + 0.160i)5-s + (−0.659 − 1.14i)7-s + (−0.0179 − 0.0122i)9-s + (1.17 − 0.567i)11-s + (0.350 + 0.0528i)13-s + (0.0127 + 0.169i)15-s + (0.0498 − 0.127i)17-s + (1.44 − 0.986i)19-s + (−0.290 − 1.27i)21-s + (−0.115 + 1.54i)23-s + (0.711 − 0.659i)25-s + (−0.630 − 0.790i)27-s + (0.605 − 0.186i)29-s + (0.128 + 0.119i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.446i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.98139 - 0.466779i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.98139 - 0.466779i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 43 | \( 1 + (3.30 - 5.66i)T \) |
| good | 3 | \( 1 + (-1.63 - 0.504i)T + (2.47 + 1.68i)T^{2} \) |
| 5 | \( 1 + (-0.140 - 0.358i)T + (-3.66 + 3.40i)T^{2} \) |
| 7 | \( 1 + (1.74 + 3.02i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.90 + 1.88i)T + (6.85 - 8.60i)T^{2} \) |
| 13 | \( 1 + (-1.26 - 0.190i)T + (12.4 + 3.83i)T^{2} \) |
| 17 | \( 1 + (-0.205 + 0.523i)T + (-12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (-6.30 + 4.29i)T + (6.94 - 17.6i)T^{2} \) |
| 23 | \( 1 + (0.553 - 7.39i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (-3.26 + 1.00i)T + (23.9 - 16.3i)T^{2} \) |
| 31 | \( 1 + (-0.717 - 0.665i)T + (2.31 + 30.9i)T^{2} \) |
| 37 | \( 1 + (2.19 - 3.80i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.07 - 4.72i)T + (-36.9 - 17.7i)T^{2} \) |
| 47 | \( 1 + (3.93 + 1.89i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (9.56 - 1.44i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (-2.92 - 3.66i)T + (-13.1 + 57.5i)T^{2} \) |
| 61 | \( 1 + (-3.96 + 3.67i)T + (4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (10.8 - 7.39i)T + (24.4 - 62.3i)T^{2} \) |
| 71 | \( 1 + (0.570 + 7.61i)T + (-70.2 + 10.5i)T^{2} \) |
| 73 | \( 1 + (2.05 + 0.309i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (4.14 + 7.17i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-11.1 - 3.43i)T + (68.5 + 46.7i)T^{2} \) |
| 89 | \( 1 + (-2.60 - 0.804i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + (-0.441 + 0.212i)T + (60.4 - 75.8i)T^{2} \) |
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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
−10.12552757444231065468386124514, −9.520958520890051077217600531554, −8.849217542853955316912094778576, −7.84938167203908980999356923911, −6.91842792804319881512879139058, −6.15531947100530503394636495473, −4.66089544108925057216002188323, −3.47968114242311114701167730726, −3.12954667573932721445779154101, −1.11501911832903540628611418960,
1.65591487973648686605005848566, 2.82556332077466180088423452418, 3.69965756699701646767848480885, 5.15853738709849745653607687391, 6.15465873384421502118131406902, 7.04990046402772205038201794032, 8.143263784071778995003982842488, 8.901488874555174050067948764044, 9.350110107557081961270985388555, 10.32387613106196454062453473924
