| L(s) = 1 | + (−0.657 − 0.292i)2-s + (−0.0382 + 0.363i)3-s + (−0.992 − 1.10i)4-s − 0.138·5-s + (0.131 − 0.227i)6-s + (−1.74 − 1.94i)7-s + (0.774 + 2.38i)8-s + (2.80 + 0.595i)9-s + (0.0907 + 0.0404i)10-s + (−2.01 + 0.427i)11-s + (0.438 − 0.318i)12-s + (−2.66 − 2.42i)13-s + (0.580 + 1.78i)14-s + (0.00527 − 0.0502i)15-s + (−0.121 + 1.15i)16-s + (−4.54 − 0.966i)17-s + ⋯ |
| L(s) = 1 | + (−0.464 − 0.206i)2-s + (−0.0220 + 0.209i)3-s + (−0.496 − 0.550i)4-s − 0.0617·5-s + (0.0536 − 0.0929i)6-s + (−0.660 − 0.733i)7-s + (0.273 + 0.842i)8-s + (0.934 + 0.198i)9-s + (0.0287 + 0.0127i)10-s + (−0.606 + 0.128i)11-s + (0.126 − 0.0919i)12-s + (−0.738 − 0.673i)13-s + (0.155 + 0.477i)14-s + (0.00136 − 0.0129i)15-s + (−0.0304 + 0.289i)16-s + (−1.10 − 0.234i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0719i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00734128 + 0.203681i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00734128 + 0.203681i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.66 + 2.42i)T \) |
| 31 | \( 1 + (1.53 + 5.35i)T \) |
| good | 2 | \( 1 + (0.657 + 0.292i)T + (1.33 + 1.48i)T^{2} \) |
| 3 | \( 1 + (0.0382 - 0.363i)T + (-2.93 - 0.623i)T^{2} \) |
| 5 | \( 1 + 0.138T + 5T^{2} \) |
| 7 | \( 1 + (1.74 + 1.94i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 + (2.01 - 0.427i)T + (10.0 - 4.47i)T^{2} \) |
| 17 | \( 1 + (4.54 + 0.966i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (0.0199 + 0.190i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (5.22 - 5.80i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.320 + 0.142i)T + (19.4 + 21.5i)T^{2} \) |
| 37 | \( 1 + (3.98 + 6.90i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.827 + 0.368i)T + (27.4 + 30.4i)T^{2} \) |
| 43 | \( 1 + (-1.06 - 10.1i)T + (-42.0 + 8.94i)T^{2} \) |
| 47 | \( 1 + (-2.99 - 2.17i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (2.65 + 8.16i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (3.95 - 1.75i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (1.43 - 2.47i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.624 - 1.08i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.2 - 2.39i)T + (64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-2.08 + 6.43i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.484 + 1.49i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.52 + 3.28i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.87 + 0.398i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (8.66 + 9.62i)T + (-10.1 + 96.4i)T^{2} \) |
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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
−10.51669706602290118939862632392, −9.851321385948597058788851592302, −9.407757153112446223694851456983, −7.963912896231246176673169205626, −7.28775390828799689070185800975, −5.91433619137760404760987731234, −4.82142801258937724064297873860, −3.85569032144630476442317965218, −2.04193612821513516961812989127, −0.14873986434345689135616879722,
2.27024762887864438346274143605, 3.80468906383099442204588026731, 4.81278865348124924748285361051, 6.35122226372544572302510854513, 7.10722868873860001241233417121, 8.127650903881803817179843618028, 8.977030988154149324769685546537, 9.736259306380458330624356087580, 10.54604313103361964558851028109, 12.17436723629743668539409316170
