| L(s) = 1 | + (0.273 − 2.60i)2-s + (−1.28 + 1.15i)3-s + (−4.74 − 1.00i)4-s + (−0.338 − 3.21i)5-s + (2.66 + 3.66i)6-s + (−0.669 + 0.743i)7-s + (−2.30 + 7.10i)8-s + (0.313 − 2.98i)9-s − 8.47·10-s + (3.19 − 0.907i)11-s + (7.28 − 4.20i)12-s + (2.25 − 1.00i)13-s + (1.75 + 1.94i)14-s + (4.16 + 3.75i)15-s + (9.00 + 4.00i)16-s + (1.5 − 1.08i)17-s + ⋯ |
| L(s) = 1 | + (0.193 − 1.84i)2-s + (−0.743 + 0.669i)3-s + (−2.37 − 0.504i)4-s + (−0.151 − 1.43i)5-s + (1.08 + 1.49i)6-s + (−0.252 + 0.280i)7-s + (−0.816 + 2.51i)8-s + (0.104 − 0.994i)9-s − 2.67·10-s + (0.961 − 0.273i)11-s + (2.10 − 1.21i)12-s + (0.626 − 0.278i)13-s + (0.468 + 0.519i)14-s + (1.07 + 0.968i)15-s + (2.25 + 1.00i)16-s + (0.363 − 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.160i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0597092 - 0.738595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0597092 - 0.738595i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.28 - 1.15i)T \) |
| 11 | \( 1 + (-3.19 + 0.907i)T \) |
| good | 2 | \( 1 + (-0.273 + 2.60i)T + (-1.95 - 0.415i)T^{2} \) |
| 5 | \( 1 + (0.338 + 3.21i)T + (-4.89 + 1.03i)T^{2} \) |
| 7 | \( 1 + (0.669 - 0.743i)T + (-0.731 - 6.96i)T^{2} \) |
| 13 | \( 1 + (-2.25 + 1.00i)T + (8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 1.08i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.309 + 0.951i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (1.19 - 2.06i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.24 + 1.37i)T + (-3.03 - 28.8i)T^{2} \) |
| 31 | \( 1 + (1.47 - 0.658i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.88 - 5.79i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (7.67 + 8.52i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.42 - 4.20i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.81 + 1.23i)T + (42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-5.73 - 4.16i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.604 - 0.128i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-3.78 - 1.68i)T + (40.8 + 45.3i)T^{2} \) |
| 67 | \( 1 + (3 - 5.19i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.3 + 8.28i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (2.11 + 6.51i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-0.999 + 9.50i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (13.0 + 5.79i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + (0.627 - 5.96i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−12.90232244860967310589335724569, −12.06126021996737734181186008482, −11.56211972424525550076978200066, −10.35473419101459390857665709353, −9.315749648940128780146688296831, −8.726205478127253526925440964613, −5.73359513960011081183564375796, −4.63612257304490819939521844948, −3.59883759348807982760420571243, −1.03052895502872335515758168140,
3.96803792175130361880542707432, 5.73600587681427051295515295388, 6.70067733735902791900135502402, 7.10226570793421977506576400702, 8.348656739378811152056245382594, 9.988626361416851284405332897846, 11.30449767826406213913986265290, 12.64104778261704215673278489942, 13.84067790784103395935898800350, 14.40579846018543957571518929096
