| L(s) = 1 | + (−4.97 + 1.33i)2-s + (73.7 − 127. i)3-s + (−198. + 114. i)4-s + (−65.5 − 65.5i)5-s + (−196. + 733. i)6-s + (−2.44e3 − 655. i)7-s + (1.76e3 − 1.76e3i)8-s + (−7.59e3 − 1.31e4i)9-s + (413. + 238. i)10-s + (−3.56e3 − 1.33e4i)11-s + 3.38e4i·12-s + (5.08e3 + 2.81e4i)13-s + 1.30e4·14-s + (−1.32e4 + 3.53e3i)15-s + (2.29e4 − 3.97e4i)16-s + (6.08e4 − 3.51e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.310 + 0.0832i)2-s + (0.910 − 1.57i)3-s + (−0.776 + 0.448i)4-s + (−0.104 − 0.104i)5-s + (−0.151 + 0.565i)6-s + (−1.01 − 0.272i)7-s + (0.431 − 0.431i)8-s + (−1.15 − 2.00i)9-s + (0.0413 + 0.0238i)10-s + (−0.243 − 0.908i)11-s + 1.63i·12-s + (0.177 + 0.984i)13-s + 0.339·14-s + (−0.260 + 0.0698i)15-s + (0.350 − 0.606i)16-s + (0.728 − 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.378813 - 0.993896i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.378813 - 0.993896i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-5.08e3 - 2.81e4i)T \) |
| good | 2 | \( 1 + (4.97 - 1.33i)T + (221. - 128i)T^{2} \) |
| 3 | \( 1 + (-73.7 + 127. i)T + (-3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (65.5 + 65.5i)T + 3.90e5iT^{2} \) |
| 7 | \( 1 + (2.44e3 + 655. i)T + (4.99e6 + 2.88e6i)T^{2} \) |
| 11 | \( 1 + (3.56e3 + 1.33e4i)T + (-1.85e8 + 1.07e8i)T^{2} \) |
| 17 | \( 1 + (-6.08e4 + 3.51e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.01e4 + 3.78e4i)T + (-1.47e10 - 8.49e9i)T^{2} \) |
| 23 | \( 1 + (-2.78e5 - 1.60e5i)T + (3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-3.48e5 + 6.03e5i)T + (-2.50e11 - 4.33e11i)T^{2} \) |
| 31 | \( 1 + (-7.04e5 - 7.04e5i)T + 8.52e11iT^{2} \) |
| 37 | \( 1 + (6.46e5 + 2.41e6i)T + (-3.04e12 + 1.75e12i)T^{2} \) |
| 41 | \( 1 + (3.75e6 - 1.00e6i)T + (6.91e12 - 3.99e12i)T^{2} \) |
| 43 | \( 1 + (-1.88e6 + 1.09e6i)T + (5.84e12 - 1.01e13i)T^{2} \) |
| 47 | \( 1 + (5.30e6 - 5.30e6i)T - 2.38e13iT^{2} \) |
| 53 | \( 1 - 1.37e5T + 6.22e13T^{2} \) |
| 59 | \( 1 + (-1.15e7 - 3.10e6i)T + (1.27e14 + 7.34e13i)T^{2} \) |
| 61 | \( 1 + (2.44e6 + 4.23e6i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (9.87e6 - 2.64e6i)T + (3.51e14 - 2.03e14i)T^{2} \) |
| 71 | \( 1 + (-9.85e5 + 3.67e6i)T + (-5.59e14 - 3.22e14i)T^{2} \) |
| 73 | \( 1 + (-3.80e7 + 3.80e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 1.02e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + (-2.14e7 - 2.14e7i)T + 2.25e15iT^{2} \) |
| 89 | \( 1 + (-2.07e7 - 7.75e7i)T + (-3.40e15 + 1.96e15i)T^{2} \) |
| 97 | \( 1 + (1.54e7 - 5.74e7i)T + (-6.78e15 - 3.91e15i)T^{2} \) |
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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
−17.82020216390745008868259752572, −16.35969603154124711691994986719, −14.00107440097339607129668484263, −13.37429230251578496928088803487, −12.18837157573579658391908686568, −9.314509316103974123350424527942, −8.142147938738917692265164967190, −6.75999003326718621801605052373, −3.22030877338622513957385093514, −0.67930255293525061680079740853,
3.31640255908189983347404615442, 5.06872600776321120454876226482, 8.424308716561544382696404085476, 9.721226040179060409517593336214, 10.36851968522165098222597201899, 13.14816293686208733403177598712, 14.72836972858875181374803110367, 15.47829513996681455342617553938, 16.95591049559621076303720440918, 18.80452733273952360571548960690
