| L(s) = 1 | + (−0.996 − 0.0871i)2-s + (1.72 + 0.0962i)3-s + (0.984 + 0.173i)4-s + (−1.16 − 1.90i)5-s + (−1.71 − 0.246i)6-s + (1.36 + 0.956i)7-s + (−0.965 − 0.258i)8-s + (2.98 + 0.332i)9-s + (0.992 + 2.00i)10-s + (1.17 − 3.23i)11-s + (1.68 + 0.395i)12-s + (−0.0285 − 0.326i)13-s + (−1.27 − 1.07i)14-s + (−1.82 − 3.41i)15-s + (0.939 + 0.342i)16-s + (2.17 − 0.583i)17-s + ⋯ |
| L(s) = 1 | + (−0.704 − 0.0616i)2-s + (0.998 + 0.0555i)3-s + (0.492 + 0.0868i)4-s + (−0.520 − 0.853i)5-s + (−0.699 − 0.100i)6-s + (0.516 + 0.361i)7-s + (−0.341 − 0.0915i)8-s + (0.993 + 0.110i)9-s + (0.313 + 0.633i)10-s + (0.355 − 0.975i)11-s + (0.486 + 0.114i)12-s + (−0.00793 − 0.0906i)13-s + (−0.341 − 0.286i)14-s + (−0.472 − 0.881i)15-s + (0.234 + 0.0855i)16-s + (0.528 − 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.851 + 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.22499 - 0.346401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.22499 - 0.346401i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.996 + 0.0871i)T \) |
| 3 | \( 1 + (-1.72 - 0.0962i)T \) |
| 5 | \( 1 + (1.16 + 1.90i)T \) |
| good | 7 | \( 1 + (-1.36 - 0.956i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.17 + 3.23i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.0285 + 0.326i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-2.17 + 0.583i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.248 - 0.143i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.780 - 1.11i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (4.61 - 3.86i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.40 + 7.98i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.38 - 5.16i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-5.63 + 6.71i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.15 - 4.61i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (4.47 - 6.39i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (8.21 - 8.21i)T - 53iT^{2} \) |
| 59 | \( 1 + (10.1 - 3.67i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.55 - 8.81i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (13.6 - 1.19i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (4.96 + 2.86i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.683 - 2.55i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (1.13 + 1.35i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (1.19 - 13.6i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-0.603 - 1.04i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-15.8 + 7.39i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.75339188971622490888319310853, −10.88296542370752843542945356718, −9.524703966181888169031917832265, −8.941377657240106173574279796927, −8.106679711939367793284979281418, −7.49098293351804490127928219583, −5.82406476335394236975104589801, −4.35669686411463127513171895640, −3.06931697742418174607141906822, −1.37294204193431584153302443564,
1.84669858690282796548627136663, 3.26231852117330459911519942528, 4.48899482663015126047195850556, 6.51973996312709903116008432952, 7.44683078134447663039052358221, 7.956585812009834529968630494114, 9.133392479617552980641151414725, 10.02444224945730609652270265114, 10.82817350447962577014114117253, 11.87541994817564350175389204098
