| L(s) = 1 | + (−8 + 13.8i)2-s + (−90.5 + 156. i)3-s + (−127. − 221. i)4-s + (−153. + 265. i)5-s + (−1.44e3 − 2.50e3i)6-s + 1.21e4·7-s + 4.09e3·8-s + (−6.54e3 − 1.13e4i)9-s + (−2.45e3 − 4.24e3i)10-s + 8.74e4·11-s + 4.63e4·12-s + (2.99e4 + 5.18e4i)13-s + (−9.71e4 + 1.68e5i)14-s + (−2.77e4 − 4.80e4i)15-s + (−3.27e4 + 5.67e4i)16-s + (1.53e5 − 2.65e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.645 + 1.11i)3-s + (−0.249 − 0.433i)4-s + (−0.109 + 0.189i)5-s + (−0.456 − 0.790i)6-s + 1.91·7-s + 0.353·8-s + (−0.332 − 0.575i)9-s + (−0.0775 − 0.134i)10-s + 1.80·11-s + 0.645·12-s + (0.290 + 0.503i)13-s + (−0.675 + 1.17i)14-s + (−0.141 − 0.245i)15-s + (−0.125 + 0.216i)16-s + (0.444 − 0.770i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.973368 + 1.40781i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.973368 + 1.40781i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (8 - 13.8i)T \) |
| 19 | \( 1 + (8.38e4 + 5.61e5i)T \) |
| good | 3 | \( 1 + (90.5 - 156. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (153. - 265. i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 - 1.21e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.74e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-2.99e4 - 5.18e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-1.53e5 + 2.65e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 23 | \( 1 + (2.91e5 + 5.04e5i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-9.90e5 - 1.71e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 - 5.90e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.27e4T + 1.29e14T^{2} \) |
| 41 | \( 1 + (5.83e6 - 1.01e7i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (1.82e6 - 3.16e6i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + (-6.79e6 - 1.17e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (4.60e7 + 7.97e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-6.58e7 + 1.13e8i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (8.63e7 + 1.49e8i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (-9.56e7 - 1.65e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (6.93e7 - 1.20e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (2.01e8 - 3.49e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (2.86e8 - 4.96e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 + 1.71e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (1.32e7 + 2.28e7i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (1.87e8 - 3.24e8i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
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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
−14.74933818534795709153325786314, −14.14627034108220792072431227270, −11.57078556649500159764466297636, −11.14484701465446701774409805786, −9.613014407602488219346461609986, −8.460362851174478850134241087322, −6.82426460741154268608166130684, −5.12661387358173003266272457633, −4.27407972342922352338792436653, −1.25857420076709903134561527464,
1.06231616955327686189597501117, 1.63305202277905913840274685594, 4.22883567571409134599118170626, 6.04175007875201320593231201330, 7.66854527419865571313348205168, 8.622153562018027665143268140326, 10.57879635395264594149351795791, 11.96315126715240450173637103365, 12.01069588646880054078944669555, 13.77175723548995471773406980009
