| L(s) = 1 | + (0.849 + 2.61i)2-s + (0.0476 − 0.757i)3-s + (−4.50 + 3.27i)4-s + (1.44 + 0.182i)5-s + (2.02 − 0.519i)6-s + (−1.39 + 2.96i)7-s + (−7.93 − 5.76i)8-s + (2.40 + 0.303i)9-s + (0.749 + 3.92i)10-s + (−0.289 − 4.59i)11-s + (2.26 + 3.56i)12-s + (−0.448 + 0.542i)13-s + (−8.93 − 1.12i)14-s + (0.206 − 1.08i)15-s + (4.89 − 15.0i)16-s + (2.16 + 4.60i)17-s + ⋯ |
| L(s) = 1 | + (0.601 + 1.84i)2-s + (0.0275 − 0.437i)3-s + (−2.25 + 1.63i)4-s + (0.645 + 0.0815i)5-s + (0.825 − 0.212i)6-s + (−0.526 + 1.11i)7-s + (−2.80 − 2.03i)8-s + (0.801 + 0.101i)9-s + (0.237 + 1.24i)10-s + (−0.0871 − 1.38i)11-s + (0.653 + 1.03i)12-s + (−0.124 + 0.150i)13-s + (−2.38 − 0.301i)14-s + (0.0534 − 0.280i)15-s + (1.22 − 3.76i)16-s + (0.525 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.801 - 0.597i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.801 - 0.597i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.436419 + 1.31632i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.436419 + 1.31632i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 151 | \( 1 + (-11.5 + 4.19i)T \) |
| good | 2 | \( 1 + (-0.849 - 2.61i)T + (-1.61 + 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.0476 + 0.757i)T + (-2.97 - 0.375i)T^{2} \) |
| 5 | \( 1 + (-1.44 - 0.182i)T + (4.84 + 1.24i)T^{2} \) |
| 7 | \( 1 + (1.39 - 2.96i)T + (-4.46 - 5.39i)T^{2} \) |
| 11 | \( 1 + (0.289 + 4.59i)T + (-10.9 + 1.37i)T^{2} \) |
| 13 | \( 1 + (0.448 - 0.542i)T + (-2.43 - 12.7i)T^{2} \) |
| 17 | \( 1 + (-2.16 - 4.60i)T + (-10.8 + 13.0i)T^{2} \) |
| 19 | \( 1 + (-4.59 + 3.33i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-2.09 + 1.51i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-3.50 + 1.38i)T + (21.1 - 19.8i)T^{2} \) |
| 31 | \( 1 + (0.950 - 0.892i)T + (1.94 - 30.9i)T^{2} \) |
| 37 | \( 1 + (3.74 - 4.53i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (3.44 - 0.885i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (2.95 + 6.28i)T + (-27.4 + 33.1i)T^{2} \) |
| 47 | \( 1 + (7.93 - 2.03i)T + (41.1 - 22.6i)T^{2} \) |
| 53 | \( 1 + (1.72 - 2.72i)T + (-22.5 - 47.9i)T^{2} \) |
| 59 | \( 1 + (-2.85 + 8.79i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (0.730 + 11.6i)T + (-60.5 + 7.64i)T^{2} \) |
| 67 | \( 1 + (4.45 - 0.562i)T + (64.8 - 16.6i)T^{2} \) |
| 71 | \( 1 + (6.09 - 12.9i)T + (-45.2 - 54.7i)T^{2} \) |
| 73 | \( 1 + (-4.13 + 8.78i)T + (-46.5 - 56.2i)T^{2} \) |
| 79 | \( 1 + (1.96 + 1.84i)T + (4.96 + 78.8i)T^{2} \) |
| 83 | \( 1 + (7.88 + 4.33i)T + (44.4 + 70.0i)T^{2} \) |
| 89 | \( 1 + (-6.13 - 3.37i)T + (47.6 + 75.1i)T^{2} \) |
| 97 | \( 1 + (9.42 - 1.19i)T + (93.9 - 24.1i)T^{2} \) |
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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
−13.54836028887661872745646114021, −12.93249156787556538848352224007, −11.98352420612573266294801192883, −9.877401818407238801640813542688, −8.839050705707618651321166609546, −8.000536726875607495136740129762, −6.68996323245046991013333050086, −6.03254152370744062325568359075, −5.09581517585492588509932583954, −3.30656824447521359658562607098,
1.48372083747406084777725157887, 3.24155691502905623958895639453, 4.35568502218082690888122499059, 5.31889641101115198890810802422, 7.22997455399488973502395119992, 9.420904302476087439690598926978, 9.947703204158636194969575746360, 10.31431402598348779401055073632, 11.69501329114303332354964462122, 12.59457854983922430126903442917
