| 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
−11.82837181537567816465445397503, −10.77042866067415553642964590104, −9.925584445156627117477242504516, −9.282914971364172443597253069307, −6.80040096594200784237003055988, −6.14887728237638864396997219300, −4.74604220813751504348682163472, −4.21471308687275266457486420857, −2.34157410175441681604267477570, −1.14748545097177556891975746311,
0.54524164419680189178056412068, 1.19461418120578821908225755270, 3.18175881261700135871155395221, 5.04331562934336621627147591273, 5.80064203519908551116195630965, 6.37391546775802628065679187464, 7.69651407017101806894082419187, 8.893888829934060143295454381522, 10.40618805607637333680396126353, 11.50454894495171039660631180378
