| L(s) = 1 | + (1.21 − 2.38i)2-s + (−1.72 − 0.0846i)3-s + (−3.02 − 4.16i)4-s + (−0.0992 + 0.626i)5-s + (−2.30 + 4.01i)6-s + (1.16 − 1.36i)7-s + (−8.31 + 1.31i)8-s + (2.98 + 0.292i)9-s + (1.37 + 0.997i)10-s + (1.80 + 2.94i)11-s + (4.88 + 7.46i)12-s + (0.481 − 6.11i)13-s + (−1.83 − 4.42i)14-s + (0.224 − 1.07i)15-s + (−3.77 + 11.6i)16-s + (0.501 − 2.08i)17-s + ⋯ |
| L(s) = 1 | + (0.858 − 1.68i)2-s + (−0.998 − 0.0488i)3-s + (−1.51 − 2.08i)4-s + (−0.0444 + 0.280i)5-s + (−0.939 + 1.64i)6-s + (0.440 − 0.515i)7-s + (−2.94 + 0.465i)8-s + (0.995 + 0.0975i)9-s + (0.434 + 0.315i)10-s + (0.544 + 0.888i)11-s + (1.40 + 2.15i)12-s + (0.133 − 1.69i)13-s + (−0.490 − 1.18i)14-s + (0.0580 − 0.277i)15-s + (−0.943 + 2.90i)16-s + (0.121 − 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.224897 - 1.11564i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.224897 - 1.11564i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.72 + 0.0846i)T \) |
| 41 | \( 1 + (-3.45 + 5.39i)T \) |
| good | 2 | \( 1 + (-1.21 + 2.38i)T + (-1.17 - 1.61i)T^{2} \) |
| 5 | \( 1 + (0.0992 - 0.626i)T + (-4.75 - 1.54i)T^{2} \) |
| 7 | \( 1 + (-1.16 + 1.36i)T + (-1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (-1.80 - 2.94i)T + (-4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (-0.481 + 6.11i)T + (-12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (-0.501 + 2.08i)T + (-15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (-0.0229 - 0.291i)T + (-18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (-2.60 - 8.02i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.356 - 1.48i)T + (-25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (2.28 - 3.13i)T + (-9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.56 - 1.86i)T + (11.4 - 35.1i)T^{2} \) |
| 43 | \( 1 + (2.08 + 1.06i)T + (25.2 + 34.7i)T^{2} \) |
| 47 | \( 1 + (1.05 - 0.901i)T + (7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (2.83 - 0.679i)T + (47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (9.33 - 3.03i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-3.43 - 6.74i)T + (-35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (-4.14 + 6.76i)T + (-30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (4.29 - 2.63i)T + (32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (-2.75 + 2.75i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.787 - 1.90i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + 13.0iT - 83T^{2} \) |
| 89 | \( 1 + (1.38 + 1.18i)T + (13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (-5.88 - 3.60i)T + (44.0 + 86.4i)T^{2} \) |
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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
−12.69994330573635242596570313683, −11.96447527443529467993201630737, −10.97011219786165731086694480718, −10.46531217879670528329034130387, −9.443470080345284808349893161073, −7.28447162110783013659257611191, −5.59357907168181978561347506453, −4.73622353946830176215589966076, −3.36173549359321213195933004798, −1.29846961203711929620092242563,
4.11688073976203275602711170959, 5.00292808151221372875279708923, 6.20205603292516584539408336674, 6.78409832065898510590236958943, 8.305246882250559100771568288383, 9.166225890263818425256159277958, 11.19159179553432067802448989331, 12.14172487712447568812714278254, 12.97978320245536761630076957536, 14.15580578477265552818412951844
