| L(s) = 1 | + (22.6 + 39.1i)2-s + (−985. − 569. i)3-s + (−1.02e3 + 1.77e3i)4-s + (1.21e4 − 7.00e3i)5-s − 5.15e4i·6-s − 9.26e4·8-s + (3.81e5 + 6.61e5i)9-s + (5.49e5 + 3.17e5i)10-s + (1.52e6 − 2.63e6i)11-s + (2.01e6 − 1.16e6i)12-s − 8.06e6i·13-s − 1.59e7·15-s + (−2.09e6 − 3.63e6i)16-s + (2.92e6 + 1.68e6i)17-s + (−1.72e7 + 2.99e7i)18-s + (1.38e7 − 7.97e6i)19-s + ⋯ |
| L(s) = 1 | + (0.353 + 0.612i)2-s + (−1.35 − 0.780i)3-s + (−0.249 + 0.433i)4-s + (0.776 − 0.448i)5-s − 1.10i·6-s − 0.353·8-s + (0.718 + 1.24i)9-s + (0.549 + 0.317i)10-s + (0.858 − 1.48i)11-s + (0.676 − 0.390i)12-s − 1.67i·13-s − 1.39·15-s + (−0.125 − 0.216i)16-s + (0.121 + 0.0699i)17-s + (−0.508 + 0.880i)18-s + (0.293 − 0.169i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.0633 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(1.836122646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.836122646\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-22.6 - 39.1i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (985. + 569. i)T + (2.65e5 + 4.60e5i)T^{2} \) |
| 5 | \( 1 + (-1.21e4 + 7.00e3i)T + (1.22e8 - 2.11e8i)T^{2} \) |
| 11 | \( 1 + (-1.52e6 + 2.63e6i)T + (-1.56e12 - 2.71e12i)T^{2} \) |
| 13 | \( 1 + 8.06e6iT - 2.32e13T^{2} \) |
| 17 | \( 1 + (-2.92e6 - 1.68e6i)T + (2.91e14 + 5.04e14i)T^{2} \) |
| 19 | \( 1 + (-1.38e7 + 7.97e6i)T + (1.10e15 - 1.91e15i)T^{2} \) |
| 23 | \( 1 + (-6.09e7 - 1.05e8i)T + (-1.09e16 + 1.89e16i)T^{2} \) |
| 29 | \( 1 - 1.10e9T + 3.53e17T^{2} \) |
| 31 | \( 1 + (-1.49e8 - 8.60e7i)T + (3.93e17 + 6.82e17i)T^{2} \) |
| 37 | \( 1 + (-8.32e8 - 1.44e9i)T + (-3.29e18 + 5.70e18i)T^{2} \) |
| 41 | \( 1 + 2.84e9iT - 2.25e19T^{2} \) |
| 43 | \( 1 - 3.52e9T + 3.99e19T^{2} \) |
| 47 | \( 1 + (-4.75e9 + 2.74e9i)T + (5.80e19 - 1.00e20i)T^{2} \) |
| 53 | \( 1 + (-3.54e9 + 6.14e9i)T + (-2.45e20 - 4.25e20i)T^{2} \) |
| 59 | \( 1 + (2.94e10 + 1.69e10i)T + (8.89e20 + 1.54e21i)T^{2} \) |
| 61 | \( 1 + (6.13e10 - 3.53e10i)T + (1.32e21 - 2.29e21i)T^{2} \) |
| 67 | \( 1 + (-5.76e10 + 9.97e10i)T + (-4.09e21 - 7.08e21i)T^{2} \) |
| 71 | \( 1 + 5.03e10T + 1.64e22T^{2} \) |
| 73 | \( 1 + (-6.57e10 - 3.79e10i)T + (1.14e22 + 1.98e22i)T^{2} \) |
| 79 | \( 1 + (1.28e11 + 2.22e11i)T + (-2.95e22 + 5.11e22i)T^{2} \) |
| 83 | \( 1 - 2.08e9iT - 1.06e23T^{2} \) |
| 89 | \( 1 + (-7.55e11 + 4.36e11i)T + (1.23e23 - 2.13e23i)T^{2} \) |
| 97 | \( 1 + 8.85e11iT - 6.93e23T^{2} \) |
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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
−11.50454894495171039660631180378, −10.40618805607637333680396126353, −8.893888829934060143295454381522, −7.69651407017101806894082419187, −6.37391546775802628065679187464, −5.80064203519908551116195630965, −5.04331562934336621627147591273, −3.18175881261700135871155395221, −1.19461418120578821908225755270, −0.54524164419680189178056412068,
1.14748545097177556891975746311, 2.34157410175441681604267477570, 4.21471308687275266457486420857, 4.74604220813751504348682163472, 6.14887728237638864396997219300, 6.80040096594200784237003055988, 9.282914971364172443597253069307, 9.925584445156627117477242504516, 10.77042866067415553642964590104, 11.82837181537567816465445397503
