| L(s) = 1 | + (0.939 + 0.342i)2-s + (0.408 + 2.31i)3-s + (0.766 + 0.642i)4-s + (−1.33 + 1.12i)5-s + (−0.408 + 2.31i)6-s + (−0.571 + 0.989i)7-s + (0.500 + 0.866i)8-s + (−2.37 + 0.863i)9-s + (−1.63 + 0.596i)10-s + (0.5 + 0.866i)11-s + (−1.17 + 2.03i)12-s + (0.721 − 4.09i)13-s + (−0.875 + 0.734i)14-s + (−3.14 − 2.63i)15-s + (0.173 + 0.984i)16-s + (0.627 + 0.228i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.235 + 1.33i)3-s + (0.383 + 0.321i)4-s + (−0.597 + 0.501i)5-s + (−0.166 + 0.944i)6-s + (−0.215 + 0.374i)7-s + (0.176 + 0.306i)8-s + (−0.790 + 0.287i)9-s + (−0.518 + 0.188i)10-s + (0.150 + 0.261i)11-s + (−0.339 + 0.587i)12-s + (0.200 − 1.13i)13-s + (−0.233 + 0.196i)14-s + (−0.811 − 0.680i)15-s + (0.0434 + 0.246i)16-s + (0.152 + 0.0553i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.670 - 0.741i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.766813 + 1.72810i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.766813 + 1.72810i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (4.34 - 0.332i)T \) |
| good | 3 | \( 1 + (-0.408 - 2.31i)T + (-2.81 + 1.02i)T^{2} \) |
| 5 | \( 1 + (1.33 - 1.12i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.571 - 0.989i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-0.721 + 4.09i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.627 - 0.228i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-2.34 - 1.96i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-8.97 + 3.26i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.813 + 1.40i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 + (1.17 + 6.63i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (5.65 - 4.74i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-0.360 + 0.131i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (2.73 + 2.29i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-7.23 - 2.63i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.51 - 5.46i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.51 + 2.37i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.827 + 0.694i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (1.71 + 9.74i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.58 - 8.96i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.54 + 13.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.02 + 5.83i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (12.3 + 4.48i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.43707076158437989217541802495, −10.60263689127225909637174188687, −9.935175156961515456672369637251, −8.786952737134978716389070421503, −7.908768158078873250923721073666, −6.72385036591188915828636072802, −5.59815304723996601283221308696, −4.57794597430545575225669269987, −3.66456978859807507159543741614, −2.82331992883929230524975703551,
1.04063288060346292292364975573, 2.41572356377917429176379413846, 3.86897540145921821643096682411, 4.87459963473324662727817782493, 6.56638448301344834571728891767, 6.75422487117175247633455221617, 8.091361394107971976527741037107, 8.720786443697569763088746016228, 10.13777556802305589979665882934, 11.25635194455719037489430560294
