| L(s) = 1 | + (0.0519 + 1.41i)2-s + (−0.602 − 1.31i)3-s + (−1.99 + 0.146i)4-s + (−1.65 − 1.90i)5-s + (1.83 − 0.919i)6-s + (−0.428 − 0.275i)7-s + (−0.311 − 2.81i)8-s + (0.587 − 0.678i)9-s + (2.61 − 2.43i)10-s + (1.58 − 0.227i)11-s + (1.39 + 2.54i)12-s + (−3.61 − 5.62i)13-s + (0.366 − 0.619i)14-s + (−1.52 + 3.33i)15-s + (3.95 − 0.585i)16-s + (−0.123 − 0.418i)17-s + ⋯ |
| L(s) = 1 | + (0.0367 + 0.999i)2-s + (−0.347 − 0.761i)3-s + (−0.997 + 0.0734i)4-s + (−0.739 − 0.853i)5-s + (0.748 − 0.375i)6-s + (−0.161 − 0.104i)7-s + (−0.110 − 0.993i)8-s + (0.195 − 0.226i)9-s + (0.825 − 0.770i)10-s + (0.477 − 0.0685i)11-s + (0.402 + 0.733i)12-s + (−1.00 − 1.56i)13-s + (0.0980 − 0.165i)14-s + (−0.392 + 0.860i)15-s + (0.989 − 0.146i)16-s + (−0.0298 − 0.101i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.351 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.566424 - 0.392271i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.566424 - 0.392271i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0519 - 1.41i)T \) |
| 23 | \( 1 + (2.68 - 3.97i)T \) |
| good | 3 | \( 1 + (0.602 + 1.31i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (1.65 + 1.90i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (0.428 + 0.275i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-1.58 + 0.227i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (3.61 + 5.62i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.123 + 0.418i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.699 - 2.38i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.17 - 4.01i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-4.20 - 1.92i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-4.26 + 4.91i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (0.744 + 0.859i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-9.04 + 4.13i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 1.35iT - 47T^{2} \) |
| 53 | \( 1 + (-4.17 - 2.68i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (10.0 - 6.42i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.54 + 7.76i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-3.09 - 0.444i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-2.09 - 0.301i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.08 - 1.78i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-13.8 + 8.91i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (8.69 + 7.53i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-14.6 + 6.69i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (2.94 - 2.54i)T + (13.8 - 96.0i)T^{2} \) |
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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
−12.46355330072038090463542495137, −12.04395746582864712657240303676, −10.24592956036410088438812884908, −9.138817218027593606152306583678, −7.962888175837830871376853641964, −7.40549764922690029680747046541, −6.17589349656865842842007471541, −5.09450133589563906057909217448, −3.79222012775718727269950146971, −0.66502764790435578502669146536,
2.47634179375894523433094694435, 4.05585353254155506469994418895, 4.66745530178124871144068362905, 6.48681173738465311493691461053, 7.82356472527425365873913912264, 9.297524798700675833284172192502, 9.964900641475803750040334452158, 11.01883211484263091122702142628, 11.56436841875415616736264275032, 12.43101077642631929922857982630
