L(s) = 1 | + (0.562 + 1.10i)2-s + (0.397 − 2.51i)3-s + (0.274 − 0.377i)4-s + (−0.841 + 2.07i)5-s + (2.99 − 0.973i)6-s + (3.42 − 0.542i)7-s + (3.01 + 0.477i)8-s + (−3.30 − 1.07i)9-s + (−2.75 + 0.235i)10-s + (−0.839 − 0.839i)12-s + (−0.658 + 0.335i)13-s + (2.52 + 3.47i)14-s + (4.87 + 2.94i)15-s + (0.880 + 2.70i)16-s + (0.550 + 0.280i)17-s + (−0.672 − 4.24i)18-s + ⋯ |
L(s) = 1 | + (0.397 + 0.780i)2-s + (0.229 − 1.45i)3-s + (0.137 − 0.188i)4-s + (−0.376 + 0.926i)5-s + (1.22 − 0.397i)6-s + (1.29 − 0.205i)7-s + (1.06 + 0.168i)8-s + (−1.10 − 0.357i)9-s + (−0.872 + 0.0744i)10-s + (−0.242 − 0.242i)12-s + (−0.182 + 0.0931i)13-s + (0.674 + 0.928i)14-s + (1.25 + 0.759i)15-s + (0.220 + 0.677i)16-s + (0.133 + 0.0679i)17-s + (−0.158 − 1.00i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.30817 - 0.304039i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.30817 - 0.304039i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.841 - 2.07i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.562 - 1.10i)T + (-1.17 + 1.61i)T^{2} \) |
| 3 | \( 1 + (-0.397 + 2.51i)T + (-2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 + (-3.42 + 0.542i)T + (6.65 - 2.16i)T^{2} \) |
| 13 | \( 1 + (0.658 - 0.335i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.550 - 0.280i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-2.54 + 1.84i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (4.30 - 4.30i)T - 23iT^{2} \) |
| 29 | \( 1 + (2.34 + 1.70i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-0.937 + 2.88i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (0.359 + 2.27i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-0.599 - 0.825i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-4.07 - 4.07i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.07 + 0.169i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (1.94 + 3.82i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (4.35 - 5.98i)T + (-18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-3.56 + 1.15i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (9.39 + 9.39i)T + 67iT^{2} \) |
| 71 | \( 1 + (-1.11 - 3.43i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.0841 - 0.531i)T + (-69.4 + 22.5i)T^{2} \) |
| 79 | \( 1 + (3.77 - 11.6i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (7.27 - 14.2i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 - 14.6iT - 89T^{2} \) |
| 97 | \( 1 + (12.7 - 6.51i)T + (57.0 - 78.4i)T^{2} \) |
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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
−10.98077888049270009962015707716, −9.772681344369154899124001385364, −8.128126974000083815837866323265, −7.71308284164140838946407342630, −7.15920124804408603703019147518, −6.31666410174356836111660134119, −5.41423356514909578918630072092, −4.14311820507561655699660321240, −2.44739340201665724056569209489, −1.42451443227436182952987286841,
1.68289942416825689906400868752, 3.14776429104531364151393677937, 4.21310990831534721536878938200, 4.66286262359050003278891115261, 5.52261568684501701430736071238, 7.49600408469225901657199567032, 8.269761199525220003701008094129, 9.005980243222074547652468687640, 10.07410008103979780379258272378, 10.71032475260488559451888795237