| L(s) = 1 | + (−1.40 + 0.738i)2-s + (−0.456 + 0.660i)3-s + (0.297 − 0.431i)4-s + (2.02 + 2.93i)5-s + (0.153 − 1.26i)6-s + (−0.0376 − 0.310i)7-s + (0.282 − 2.32i)8-s + (0.835 + 2.20i)9-s + (−5.02 − 2.63i)10-s + (−0.307 − 2.53i)11-s + (0.149 + 0.393i)12-s − 5.98·13-s + (0.281 + 0.408i)14-s − 2.86·15-s + (1.69 + 4.46i)16-s + (0.868 − 0.214i)17-s + ⋯ |
| L(s) = 1 | + (−0.994 + 0.522i)2-s + (−0.263 + 0.381i)3-s + (0.148 − 0.215i)4-s + (0.906 + 1.31i)5-s + (0.0627 − 0.516i)6-s + (−0.0142 − 0.117i)7-s + (0.0999 − 0.822i)8-s + (0.278 + 0.734i)9-s + (−1.58 − 0.833i)10-s + (−0.0926 − 0.762i)11-s + (0.0430 + 0.113i)12-s − 1.66·13-s + (0.0753 + 0.109i)14-s − 0.740·15-s + (0.423 + 1.11i)16-s + (0.210 − 0.0519i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.160599 - 0.324326i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.160599 - 0.324326i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 859 | \( 1 + (14.5 + 25.4i)T \) |
| good | 2 | \( 1 + (1.40 - 0.738i)T + (1.13 - 1.64i)T^{2} \) |
| 3 | \( 1 + (0.456 - 0.660i)T + (-1.06 - 2.80i)T^{2} \) |
| 5 | \( 1 + (-2.02 - 2.93i)T + (-1.77 + 4.67i)T^{2} \) |
| 7 | \( 1 + (0.0376 + 0.310i)T + (-6.79 + 1.67i)T^{2} \) |
| 11 | \( 1 + (0.307 + 2.53i)T + (-10.6 + 2.63i)T^{2} \) |
| 13 | \( 1 + 5.98T + 13T^{2} \) |
| 17 | \( 1 + (-0.868 + 0.214i)T + (15.0 - 7.90i)T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 + (3.67 + 1.92i)T + (13.0 + 18.9i)T^{2} \) |
| 29 | \( 1 + (2.51 - 6.64i)T + (-21.7 - 19.2i)T^{2} \) |
| 31 | \( 1 + (4.92 + 4.36i)T + (3.73 + 30.7i)T^{2} \) |
| 37 | \( 1 + (-4.13 - 10.9i)T + (-27.6 + 24.5i)T^{2} \) |
| 41 | \( 1 + (-5.75 + 8.33i)T + (-14.5 - 38.3i)T^{2} \) |
| 43 | \( 1 - 0.477T + 43T^{2} \) |
| 47 | \( 1 + (8.16 - 7.23i)T + (5.66 - 46.6i)T^{2} \) |
| 53 | \( 1 + (8.26 + 2.03i)T + (46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (12.2 - 6.45i)T + (33.5 - 48.5i)T^{2} \) |
| 61 | \( 1 - 0.280T + 61T^{2} \) |
| 67 | \( 1 + (4.22 + 11.1i)T + (-50.1 + 44.4i)T^{2} \) |
| 71 | \( 1 + (-5.60 - 1.38i)T + (62.8 + 32.9i)T^{2} \) |
| 73 | \( 1 + (-0.414 + 3.41i)T + (-70.8 - 17.4i)T^{2} \) |
| 79 | \( 1 + (-7.22 + 10.4i)T + (-28.0 - 73.8i)T^{2} \) |
| 83 | \( 1 + (-0.787 - 6.48i)T + (-80.5 + 19.8i)T^{2} \) |
| 89 | \( 1 + (-5.73 - 1.41i)T + (78.8 + 41.3i)T^{2} \) |
| 97 | \( 1 + (4.87 - 1.20i)T + (85.8 - 45.0i)T^{2} \) |
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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
−10.62606486360732362376897633416, −9.754648216700329988319208597491, −9.332663943049301965575080668482, −7.979975152453155910315389498621, −7.43611603402845617255375945563, −6.58723711380528495246912858216, −5.77262299373342861926362257002, −4.59910267570431737144811819673, −3.21549357250001598143448883604, −2.05071293606146339568386618847,
0.24319557691602951486927728392, 1.56969818251543552743199215819, 2.30413290345589963447810268740, 4.36003661055785373459067945195, 5.25462088045070927058100357096, 6.02012844542413268028716133742, 7.33652367178091364319306541555, 8.084816780482347999616581606221, 9.279751730892419899379246630331, 9.505688381171363861899651480202
