| L(s) = 1 | + (−0.911 + 0.345i)2-s + (0.568 − 0.822i)3-s + (−0.785 + 0.695i)4-s + (2.27 − 0.276i)5-s + (−0.233 + 0.946i)6-s + (1.78 − 3.39i)7-s + (1.38 − 2.63i)8-s + (−0.354 − 0.935i)9-s + (−1.98 + 1.04i)10-s + (−0.0539 − 0.0204i)11-s + (0.126 + 1.04i)12-s + (−1.20 + 3.39i)13-s + (−0.450 + 3.71i)14-s + (1.06 − 2.03i)15-s + (−0.0965 + 0.795i)16-s + (−4.90 − 2.57i)17-s + ⋯ |
| L(s) = 1 | + (−0.644 + 0.244i)2-s + (0.327 − 0.475i)3-s + (−0.392 + 0.347i)4-s + (1.01 − 0.123i)5-s + (−0.0952 + 0.386i)6-s + (0.673 − 1.28i)7-s + (0.488 − 0.930i)8-s + (−0.118 − 0.311i)9-s + (−0.626 + 0.329i)10-s + (−0.0162 − 0.00616i)11-s + (0.0365 + 0.300i)12-s + (−0.334 + 0.942i)13-s + (−0.120 + 0.992i)14-s + (0.275 − 0.524i)15-s + (−0.0241 + 0.198i)16-s + (−1.19 − 0.624i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.07168 - 0.546638i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.07168 - 0.546638i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.568 + 0.822i)T \) |
| 13 | \( 1 + (1.20 - 3.39i)T \) |
| good | 2 | \( 1 + (0.911 - 0.345i)T + (1.49 - 1.32i)T^{2} \) |
| 5 | \( 1 + (-2.27 + 0.276i)T + (4.85 - 1.19i)T^{2} \) |
| 7 | \( 1 + (-1.78 + 3.39i)T + (-3.97 - 5.76i)T^{2} \) |
| 11 | \( 1 + (0.0539 + 0.0204i)T + (8.23 + 7.29i)T^{2} \) |
| 17 | \( 1 + (4.90 + 2.57i)T + (9.65 + 13.9i)T^{2} \) |
| 19 | \( 1 + 6.27iT - 19T^{2} \) |
| 23 | \( 1 - 3.51T + 23T^{2} \) |
| 29 | \( 1 + (-2.75 - 7.25i)T + (-21.7 + 19.2i)T^{2} \) |
| 31 | \( 1 + (-1.70 + 6.89i)T + (-27.4 - 14.4i)T^{2} \) |
| 37 | \( 1 + (-1.92 + 7.80i)T + (-32.7 - 17.1i)T^{2} \) |
| 41 | \( 1 + (-8.26 - 5.70i)T + (14.5 + 38.3i)T^{2} \) |
| 43 | \( 1 + (1.62 - 0.399i)T + (38.0 - 19.9i)T^{2} \) |
| 47 | \( 1 + (-2.41 + 2.73i)T + (-5.66 - 46.6i)T^{2} \) |
| 53 | \( 1 + (3.03 + 1.59i)T + (30.1 + 43.6i)T^{2} \) |
| 59 | \( 1 + (3.83 - 0.465i)T + (57.2 - 14.1i)T^{2} \) |
| 61 | \( 1 + (-10.9 + 5.77i)T + (34.6 - 50.2i)T^{2} \) |
| 67 | \( 1 + (2.39 - 2.69i)T + (-8.07 - 66.5i)T^{2} \) |
| 71 | \( 1 + (-7.11 - 4.91i)T + (25.1 + 66.3i)T^{2} \) |
| 73 | \( 1 + (9.89 + 3.75i)T + (54.6 + 48.4i)T^{2} \) |
| 79 | \( 1 + (-10.5 - 9.34i)T + (9.52 + 78.4i)T^{2} \) |
| 83 | \( 1 + (2.75 - 1.90i)T + (29.4 - 77.6i)T^{2} \) |
| 89 | \( 1 - 15.4iT - 89T^{2} \) |
| 97 | \( 1 + (-3.45 - 0.419i)T + (94.1 + 23.2i)T^{2} \) |
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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
−10.69656488836913652445750310428, −9.359911508989911744003521438964, −9.267791518361391001790883338731, −8.083825418151550231268191007039, −7.12214928972793931981846650723, −6.71345704802214361658997294162, −4.94334968036212995995576729040, −4.13496231787723446431553123865, −2.38966047987403172878257079421, −0.926747852198566053618770005335,
1.74434280673400993510686248702, 2.65986402454920301192227468789, 4.52812008040512289362533916903, 5.47340818453363428179015288452, 6.10778845790477044548611274916, 7.968465693497641748204455840649, 8.579574891669162829524885053473, 9.295579258993439157169738893035, 10.12172601321466082275951465265, 10.62872720532705546412768816241
