| L(s) = 1 | + (0.0447 + 0.0131i)3-s + (0.841 − 0.540i)5-s + (0.423 − 2.94i)7-s + (−2.52 − 1.62i)9-s + (0.116 + 0.254i)11-s + (−0.287 − 1.99i)13-s + (0.0447 − 0.0131i)15-s + (1.46 − 1.69i)17-s + (−2.64 − 3.04i)19-s + (0.0577 − 0.126i)21-s + (2.21 + 4.25i)23-s + (0.415 − 0.909i)25-s + (−0.183 − 0.211i)27-s + (4.42 − 5.10i)29-s + (−3.52 + 1.03i)31-s + ⋯ |
| L(s) = 1 | + (0.0258 + 0.00758i)3-s + (0.376 − 0.241i)5-s + (0.160 − 1.11i)7-s + (−0.840 − 0.540i)9-s + (0.0349 + 0.0766i)11-s + (−0.0796 − 0.553i)13-s + (0.0115 − 0.00339i)15-s + (0.356 − 0.411i)17-s + (−0.606 − 0.699i)19-s + (0.0125 − 0.0275i)21-s + (0.462 + 0.886i)23-s + (0.0830 − 0.181i)25-s + (−0.0352 − 0.0407i)27-s + (0.821 − 0.947i)29-s + (−0.632 + 0.185i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.04055 - 0.833717i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.04055 - 0.833717i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.841 + 0.540i)T \) |
| 23 | \( 1 + (-2.21 - 4.25i)T \) |
| good | 3 | \( 1 + (-0.0447 - 0.0131i)T + (2.52 + 1.62i)T^{2} \) |
| 7 | \( 1 + (-0.423 + 2.94i)T + (-6.71 - 1.97i)T^{2} \) |
| 11 | \( 1 + (-0.116 - 0.254i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (0.287 + 1.99i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (-1.46 + 1.69i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (2.64 + 3.04i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-4.42 + 5.10i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (3.52 - 1.03i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-5.26 - 3.38i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-7.90 + 5.08i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-1.04 - 0.306i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 4.44T + 47T^{2} \) |
| 53 | \( 1 + (0.0276 - 0.192i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (1.27 + 8.87i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (2.28 - 0.670i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (4.17 - 9.14i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (5.39 - 11.8i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-8.19 - 9.45i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.917 + 6.37i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-7.09 - 4.55i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-0.984 - 0.289i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (10.8 - 6.94i)T + (40.2 - 88.2i)T^{2} \) |
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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
−10.89569166169854068405609392474, −9.940626427944202013189323365190, −9.149223177112084414228294812502, −8.126927699989528097549670221934, −7.20816807348607528326148663036, −6.16845089227265526068085795009, −5.13504673571800032785223623566, −3.97921567671417376860503855551, −2.72922462932695997408677178541, −0.844313819047024658433056171785,
2.00413836661084290101875823207, 3.01243130770561850171266073086, 4.62813799732868835268475051345, 5.72050680805973398360903989124, 6.35755843276970517112884144129, 7.74915069311706280034305672804, 8.648818550462587075142055906001, 9.280494065924265469202882671280, 10.50262221984742936829159159495, 11.16315105526023383932572494748
