| L(s) = 1 | + (−0.378 − 1.36i)2-s + (−1.15 + 0.805i)3-s + (−1.71 + 1.03i)4-s + (2.01 − 0.962i)5-s + (1.53 + 1.26i)6-s + (−0.399 − 0.107i)7-s + (2.05 + 1.94i)8-s + (−0.351 + 0.966i)9-s + (−2.07 − 2.38i)10-s + (−1.78 − 1.03i)11-s + (1.13 − 2.56i)12-s + (3.93 + 2.75i)13-s + (0.00541 + 0.585i)14-s + (−1.54 + 2.73i)15-s + (1.87 − 3.53i)16-s + (1.38 + 0.647i)17-s + ⋯ |
| L(s) = 1 | + (−0.267 − 0.963i)2-s + (−0.663 + 0.464i)3-s + (−0.856 + 0.515i)4-s + (0.902 − 0.430i)5-s + (0.625 + 0.515i)6-s + (−0.151 − 0.0404i)7-s + (0.726 + 0.687i)8-s + (−0.117 + 0.322i)9-s + (−0.656 − 0.754i)10-s + (−0.538 − 0.310i)11-s + (0.328 − 0.740i)12-s + (1.09 + 0.763i)13-s + (0.00144 + 0.156i)14-s + (−0.399 + 0.705i)15-s + (0.467 − 0.883i)16-s + (0.337 + 0.157i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00183 - 0.195291i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00183 - 0.195291i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.378 + 1.36i)T \) |
| 5 | \( 1 + (-2.01 + 0.962i)T \) |
| 19 | \( 1 + (-2.00 - 3.86i)T \) |
| good | 3 | \( 1 + (1.15 - 0.805i)T + (1.02 - 2.81i)T^{2} \) |
| 7 | \( 1 + (0.399 + 0.107i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (1.78 + 1.03i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.93 - 2.75i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-1.38 - 0.647i)T + (10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-9.37 - 0.819i)T + (22.6 + 3.99i)T^{2} \) |
| 29 | \( 1 + (-3.40 + 9.34i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (3.21 - 1.85i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.11 - 3.11i)T + 37iT^{2} \) |
| 41 | \( 1 + (1.07 - 6.09i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.545 - 6.23i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-6.51 + 3.03i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (0.907 - 10.3i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-1.37 + 0.500i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.108 + 0.0907i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (1.19 - 0.557i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (3.88 - 4.62i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (5.62 + 8.03i)T + (-24.9 + 68.5i)T^{2} \) |
| 79 | \( 1 + (0.156 - 0.886i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.66 + 9.92i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (14.5 - 2.57i)T + (83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-2.53 - 1.18i)T + (62.3 + 74.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.17538289992756640489390681936, −10.44800810391926049474078915274, −9.712050135220040025454145937270, −8.844137421419372255254891100341, −7.905077491508219260250051449950, −6.17957237442540519888673480180, −5.29230206974007260732214296728, −4.33872689933085162135949036872, −2.87080240333026839043663108805, −1.30874442394075437257367025187,
1.03244313328375566003955113981, 3.19203155735046494272116173252, 5.17725759418640850441984361317, 5.68686738803994173500819604455, 6.76061243009518644472628064092, 7.23850229500178612533930588361, 8.712359166469464589173870613962, 9.368494958921674021596930783110, 10.54727213276220303381540043578, 11.08184877863822447661858125962
