L(s) = 1 | + (2.16 − 0.581i)2-s + (2.63 − 1.52i)4-s + (−1.87 + 1.87i)5-s + (1.25 + 2.32i)7-s + (1.65 − 1.65i)8-s + (−2.98 + 5.16i)10-s + (1.20 + 4.51i)11-s + (1.52 − 3.26i)13-s + (4.07 + 4.31i)14-s + (−0.417 + 0.723i)16-s + (2.18 + 3.78i)17-s + (0.194 + 0.0521i)19-s + (−2.09 + 7.80i)20-s + (5.24 + 9.08i)22-s + (−7.20 − 4.15i)23-s + ⋯ |
L(s) = 1 | + (1.53 − 0.410i)2-s + (1.31 − 0.760i)4-s + (−0.839 + 0.839i)5-s + (0.475 + 0.879i)7-s + (0.584 − 0.584i)8-s + (−0.942 + 1.63i)10-s + (0.364 + 1.36i)11-s + (0.423 − 0.906i)13-s + (1.09 + 1.15i)14-s + (−0.104 + 0.180i)16-s + (0.530 + 0.918i)17-s + (0.0446 + 0.0119i)19-s + (−0.467 + 1.74i)20-s + (1.11 + 1.93i)22-s + (−1.50 − 0.867i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.09394 + 0.970441i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.09394 + 0.970441i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-1.25 - 2.32i)T \) |
| 13 | \( 1 + (-1.52 + 3.26i)T \) |
good | 2 | \( 1 + (-2.16 + 0.581i)T + (1.73 - i)T^{2} \) |
| 5 | \( 1 + (1.87 - 1.87i)T - 5iT^{2} \) |
| 11 | \( 1 + (-1.20 - 4.51i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (-2.18 - 3.78i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.194 - 0.0521i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (7.20 + 4.15i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-5.20 + 9.01i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.75 + 6.75i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.22 - 4.58i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (-0.136 - 0.508i)T + (-35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-2.49 + 1.44i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.928 - 0.928i)T + 47iT^{2} \) |
| 53 | \( 1 + 1.95T + 53T^{2} \) |
| 59 | \( 1 + (-1.76 + 6.57i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (2.13 - 1.23i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.40 - 0.377i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.44 + 5.37i)T + (-61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (8.75 + 8.75i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.46T + 79T^{2} \) |
| 83 | \( 1 + (5.42 - 5.42i)T - 83iT^{2} \) |
| 89 | \( 1 + (-15.0 + 4.03i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-2.84 - 0.763i)T + (84.0 + 48.5i)T^{2} \) |
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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
−10.56090643979958197825095805098, −9.877799955221339866306250765518, −8.291435070264147126926295313353, −7.75497868605692977389114239263, −6.37842088491422727528880106152, −5.92945908824529422238837057635, −4.61787628570704455191294707501, −4.05485682068710328444010627820, −2.94560568281425338865301018651, −2.06384133522884237551533557441,
1.07474226926160245315085540075, 3.19759885934890082751793539493, 3.98464606094570704303777119620, 4.63773115945402899168146403802, 5.52187418782898457539581039897, 6.52438098894347933807446943838, 7.37685290790623670455190496145, 8.255551112204291061031372897320, 9.053894585386861030056548542063, 10.42129368684708096116258645053