L(s) = 1 | + (−24.1 + 24.1i)2-s + (21.4 − 138. i)3-s − 652. i·4-s + (844. + 1.11e3i)5-s + (2.82e3 + 3.86e3i)6-s + (1.78e3 + 1.78e3i)7-s + (3.38e3 + 3.38e3i)8-s + (−1.87e4 − 5.95e3i)9-s + (−4.72e4 − 6.47e3i)10-s + 2.05e4i·11-s + (−9.04e4 − 1.40e4i)12-s + (−1.05e5 + 1.05e5i)13-s − 8.59e4·14-s + (1.72e5 − 9.32e4i)15-s + 1.70e5·16-s + (−2.69e5 + 2.69e5i)17-s + ⋯ |
L(s) = 1 | + (−1.06 + 1.06i)2-s + (0.153 − 0.988i)3-s − 1.27i·4-s + (0.604 + 0.796i)5-s + (0.890 + 1.21i)6-s + (0.280 + 0.280i)7-s + (0.291 + 0.291i)8-s + (−0.953 − 0.302i)9-s + (−1.49 − 0.204i)10-s + 0.423i·11-s + (−1.25 − 0.195i)12-s + (−1.02 + 1.02i)13-s − 0.598·14-s + (0.879 − 0.475i)15-s + 0.651·16-s + (−0.782 + 0.782i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.175803 + 0.658482i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.175803 + 0.658482i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-21.4 + 138. i)T \) |
| 5 | \( 1 + (-844. - 1.11e3i)T \) |
good | 2 | \( 1 + (24.1 - 24.1i)T - 512iT^{2} \) |
| 7 | \( 1 + (-1.78e3 - 1.78e3i)T + 4.03e7iT^{2} \) |
| 11 | \( 1 - 2.05e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 + (1.05e5 - 1.05e5i)T - 1.06e10iT^{2} \) |
| 17 | \( 1 + (2.69e5 - 2.69e5i)T - 1.18e11iT^{2} \) |
| 19 | \( 1 - 7.79e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + (7.58e5 + 7.58e5i)T + 1.80e12iT^{2} \) |
| 29 | \( 1 - 3.30e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.40e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (9.21e6 + 9.21e6i)T + 1.29e14iT^{2} \) |
| 41 | \( 1 + 2.96e6iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (2.26e7 - 2.26e7i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + (-2.16e7 + 2.16e7i)T - 1.11e15iT^{2} \) |
| 53 | \( 1 + (-5.81e7 - 5.81e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 - 5.01e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.29e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + (-4.93e7 - 4.93e7i)T + 2.72e16iT^{2} \) |
| 71 | \( 1 - 1.66e8iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-1.46e8 + 1.46e8i)T - 5.88e16iT^{2} \) |
| 79 | \( 1 + 2.00e7iT - 1.19e17T^{2} \) |
| 83 | \( 1 + (3.15e8 + 3.15e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 - 3.21e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + (1.74e8 + 1.74e8i)T + 7.60e17iT^{2} \) |
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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
−17.76305988889276588001392947389, −16.85963613389077333832396040209, −15.00594663095885360535439923149, −14.10969190576423341157103086806, −12.17041378478070127078676122523, −10.07553750107262652591273860197, −8.542535016040082900586574334604, −7.16003836839926310324721996124, −6.15873595731142437542744719997, −1.96925725969678380237089420947,
0.47642119249524970829510633015, 2.64809933856417760551575611059, 4.99916388625302725367988532408, 8.385930468607201102057510289874, 9.499894478377803556408442743554, 10.48204039999099315991773267431, 11.84964449009970853659579636992, 13.68175207036207070878950552012, 15.56203446069722194200074550470, 17.12597903216181049560715945744