L(s) = 1 | + (−141. + 213. i)2-s − 3.78e3i·3-s + (−2.56e4 − 6.02e4i)4-s + 1.10e5·5-s + (8.09e5 + 5.34e5i)6-s − 7.35e6i·7-s + (1.65e7 + 3.02e6i)8-s − 1.43e7·9-s + (−1.55e7 + 2.35e7i)10-s + 2.79e8i·11-s + (−2.28e8 + 9.73e7i)12-s + 2.85e7·13-s + (1.57e9 + 1.03e9i)14-s − 4.17e8i·15-s + (−2.97e9 + 3.09e9i)16-s − 6.15e9·17-s + ⋯ |
L(s) = 1 | + (−0.551 + 0.834i)2-s − 0.577i·3-s + (−0.392 − 0.919i)4-s + 0.281·5-s + (0.481 + 0.318i)6-s − 1.27i·7-s + (0.983 + 0.180i)8-s − 0.333·9-s + (−0.155 + 0.235i)10-s + 1.30i·11-s + (−0.531 + 0.226i)12-s + 0.0350·13-s + (1.06 + 0.703i)14-s − 0.162i·15-s + (−0.692 + 0.721i)16-s − 0.881·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{17}{2})\) |
\(\approx\) |
\(0.0522929 - 0.256088i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0522929 - 0.256088i\) |
\(L(9)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (141. - 213. i)T \) |
| 3 | \( 1 + 3.78e3iT \) |
good | 5 | \( 1 - 1.10e5T + 1.52e11T^{2} \) |
| 7 | \( 1 + 7.35e6iT - 3.32e13T^{2} \) |
| 11 | \( 1 - 2.79e8iT - 4.59e16T^{2} \) |
| 13 | \( 1 - 2.85e7T + 6.65e17T^{2} \) |
| 17 | \( 1 + 6.15e9T + 4.86e19T^{2} \) |
| 19 | \( 1 + 2.23e10iT - 2.88e20T^{2} \) |
| 23 | \( 1 + 1.78e10iT - 6.13e21T^{2} \) |
| 29 | \( 1 + 4.59e11T + 2.50e23T^{2} \) |
| 31 | \( 1 - 5.70e11iT - 7.27e23T^{2} \) |
| 37 | \( 1 + 6.55e12T + 1.23e25T^{2} \) |
| 41 | \( 1 - 6.32e12T + 6.37e25T^{2} \) |
| 43 | \( 1 - 1.18e13iT - 1.36e26T^{2} \) |
| 47 | \( 1 - 2.75e13iT - 5.66e26T^{2} \) |
| 53 | \( 1 + 1.04e14T + 3.87e27T^{2} \) |
| 59 | \( 1 + 1.51e14iT - 2.15e28T^{2} \) |
| 61 | \( 1 + 2.11e14T + 3.67e28T^{2} \) |
| 67 | \( 1 - 3.36e14iT - 1.64e29T^{2} \) |
| 71 | \( 1 + 2.76e14iT - 4.16e29T^{2} \) |
| 73 | \( 1 - 9.79e14T + 6.50e29T^{2} \) |
| 79 | \( 1 + 5.71e14iT - 2.30e30T^{2} \) |
| 83 | \( 1 + 3.91e15iT - 5.07e30T^{2} \) |
| 89 | \( 1 - 2.27e15T + 1.54e31T^{2} \) |
| 97 | \( 1 + 1.11e16T + 6.14e31T^{2} \) |
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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
−15.66652823432815135262648610227, −14.18877754967959204713699225371, −13.09355428995520038384075876613, −10.82015235494292259588077855680, −9.379010056203904285171411085002, −7.56866197135839295960833601998, −6.65162500999674745979240990012, −4.64046886754389004867515509555, −1.68541957405731364780160598864, −0.11104524376596599128944118466,
2.04935157563990595325427594418, 3.58331158377942704938214291812, 5.66989325677024470593765192691, 8.369017977414930818132044059258, 9.402168865281929278808306140254, 10.91353426323243800644726200298, 12.11826612372035170035633862724, 13.75258501189003432755880797681, 15.60298074888930185005860643638, 16.89533769319960491621028441634