L(s) = 1 | + (−450. − 242. i)2-s − 1.13e4i·3-s + (1.44e5 + 2.18e5i)4-s − 7.80e5·5-s + (−2.75e6 + 5.12e6i)6-s + 2.45e7i·7-s + (−1.20e7 − 1.33e8i)8-s − 1.29e8·9-s + (3.52e8 + 1.89e8i)10-s − 5.20e8i·11-s + (2.48e9 − 1.64e9i)12-s + 1.83e10·13-s + (5.95e9 − 1.10e10i)14-s + 8.87e9i·15-s + (−2.69e10 + 6.31e10i)16-s + 3.86e10·17-s + ⋯ |
L(s) = 1 | + (−0.880 − 0.473i)2-s − 0.577i·3-s + (0.551 + 0.834i)4-s − 0.399·5-s + (−0.273 + 0.508i)6-s + 0.608i·7-s + (−0.0899 − 0.995i)8-s − 0.333·9-s + (0.352 + 0.189i)10-s − 0.220i·11-s + (0.481 − 0.318i)12-s + 1.73·13-s + (0.288 − 0.535i)14-s + 0.230i·15-s + (−0.392 + 0.919i)16-s + 0.325·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{19}{2})\) |
\(\approx\) |
\(0.7535203289\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7535203289\) |
\(L(10)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (450. + 242. i)T \) |
| 3 | \( 1 + 1.13e4iT \) |
good | 5 | \( 1 + 7.80e5T + 3.81e12T^{2} \) |
| 7 | \( 1 - 2.45e7iT - 1.62e15T^{2} \) |
| 11 | \( 1 + 5.20e8iT - 5.55e18T^{2} \) |
| 13 | \( 1 - 1.83e10T + 1.12e20T^{2} \) |
| 17 | \( 1 - 3.86e10T + 1.40e22T^{2} \) |
| 19 | \( 1 + 8.98e10iT - 1.04e23T^{2} \) |
| 23 | \( 1 + 1.99e12iT - 3.24e24T^{2} \) |
| 29 | \( 1 + 1.93e13T + 2.10e26T^{2} \) |
| 31 | \( 1 + 1.86e13iT - 6.99e26T^{2} \) |
| 37 | \( 1 + 9.76e13T + 1.68e28T^{2} \) |
| 41 | \( 1 + 1.26e14T + 1.07e29T^{2} \) |
| 43 | \( 1 - 4.41e14iT - 2.52e29T^{2} \) |
| 47 | \( 1 + 1.38e15iT - 1.25e30T^{2} \) |
| 53 | \( 1 - 1.00e15T + 1.08e31T^{2} \) |
| 59 | \( 1 + 1.06e16iT - 7.50e31T^{2} \) |
| 61 | \( 1 + 2.00e16T + 1.36e32T^{2} \) |
| 67 | \( 1 + 5.21e16iT - 7.40e32T^{2} \) |
| 71 | \( 1 + 6.60e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 + 7.78e14T + 3.46e33T^{2} \) |
| 79 | \( 1 - 5.19e15iT - 1.43e34T^{2} \) |
| 83 | \( 1 + 4.99e16iT - 3.49e34T^{2} \) |
| 89 | \( 1 - 2.80e17T + 1.22e35T^{2} \) |
| 97 | \( 1 - 3.77e17T + 5.77e35T^{2} \) |
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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
−15.44722245667614977895288607948, −13.33789843429241489855775486225, −11.99371901122631762009477943321, −10.89766536626336160362523148838, −8.993168447944542059753626991013, −7.911111245391843001218365210950, −6.23813310620557043452688475327, −3.50155098380926206264942170601, −1.85073260369454570068620392414, −0.36754790074485132761018953840,
1.32095800435588858057878595481, 3.74474991282196542436688938599, 5.75708897925999851451328569737, 7.47724667161335808260004886327, 8.865787081427954403836392138784, 10.31520474836808229051743599641, 11.43133134312995181201538123848, 13.80544010641625929288334041911, 15.33827953115205132611273868682, 16.21595766036078595484722430102