L(s) = 1 | + (−9.29 + 4.47i)2-s + (−16.9 − 21.1i)3-s + (46.4 − 58.2i)4-s + (−29.3 + 14.1i)5-s + (252. + 121. i)6-s + (−125. − 157. i)7-s + (−97.5 + 427. i)8-s + (−109. + 479. i)9-s + (209. − 262. i)10-s + (28.6 + 125. i)11-s − 2.01e3·12-s + (52.5 + 230. i)13-s + (1.87e3 + 904. i)14-s + (794. + 382. i)15-s + (−476. − 2.08e3i)16-s + 1.59e3·17-s + ⋯ |
L(s) = 1 | + (−1.64 + 0.791i)2-s + (−1.08 − 1.35i)3-s + (1.45 − 1.81i)4-s + (−0.524 + 0.252i)5-s + (2.85 + 1.37i)6-s + (−0.971 − 1.21i)7-s + (−0.538 + 2.36i)8-s + (−0.450 + 1.97i)9-s + (0.661 − 0.829i)10-s + (0.0714 + 0.313i)11-s − 4.04·12-s + (0.0862 + 0.378i)13-s + (2.56 + 1.23i)14-s + (0.911 + 0.439i)15-s + (−0.465 − 2.03i)16-s + 1.33·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0516 - 0.998i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0516 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.0815393 + 0.0858669i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0815393 + 0.0858669i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 29 | \( 1 + (-2.54e3 - 3.74e3i)T \) |
good | 2 | \( 1 + (9.29 - 4.47i)T + (19.9 - 25.0i)T^{2} \) |
| 3 | \( 1 + (16.9 + 21.1i)T + (-54.0 + 236. i)T^{2} \) |
| 5 | \( 1 + (29.3 - 14.1i)T + (1.94e3 - 2.44e3i)T^{2} \) |
| 7 | \( 1 + (125. + 157. i)T + (-3.73e3 + 1.63e4i)T^{2} \) |
| 11 | \( 1 + (-28.6 - 125. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-52.5 - 230. i)T + (-3.34e5 + 1.61e5i)T^{2} \) |
| 17 | \( 1 - 1.59e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + (-455. + 570. i)T + (-5.50e5 - 2.41e6i)T^{2} \) |
| 23 | \( 1 + (3.24e3 + 1.56e3i)T + (4.01e6 + 5.03e6i)T^{2} \) |
| 31 | \( 1 + (-2.32e3 + 1.12e3i)T + (1.78e7 - 2.23e7i)T^{2} \) |
| 37 | \( 1 + (-920. + 4.03e3i)T + (-6.24e7 - 3.00e7i)T^{2} \) |
| 41 | \( 1 + 6.00e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (754. + 363. i)T + (9.16e7 + 1.14e8i)T^{2} \) |
| 47 | \( 1 + (-6.45e3 - 2.83e4i)T + (-2.06e8 + 9.95e7i)T^{2} \) |
| 53 | \( 1 + (1.07e4 - 5.18e3i)T + (2.60e8 - 3.26e8i)T^{2} \) |
| 59 | \( 1 + 2.33e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + (-4.69e3 - 5.88e3i)T + (-1.87e8 + 8.23e8i)T^{2} \) |
| 67 | \( 1 + (-4.22e3 + 1.85e4i)T + (-1.21e9 - 5.85e8i)T^{2} \) |
| 71 | \( 1 + (-3.91e3 - 1.71e4i)T + (-1.62e9 + 7.82e8i)T^{2} \) |
| 73 | \( 1 + (7.56e3 + 3.64e3i)T + (1.29e9 + 1.62e9i)T^{2} \) |
| 79 | \( 1 + (8.55e3 - 3.74e4i)T + (-2.77e9 - 1.33e9i)T^{2} \) |
| 83 | \( 1 + (4.74e4 - 5.95e4i)T + (-8.76e8 - 3.84e9i)T^{2} \) |
| 89 | \( 1 + (-6.27e3 + 3.01e3i)T + (3.48e9 - 4.36e9i)T^{2} \) |
| 97 | \( 1 + (-9.62e3 + 1.20e4i)T + (-1.91e9 - 8.37e9i)T^{2} \) |
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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
−16.72265636985233146775901774119, −15.98946223884902685216143022756, −14.02681003364583300873090509424, −12.35091329376019436484406631548, −11.01451770952650539098502130905, −9.910979188133131819127205203362, −7.80390493755870342174609605998, −7.10089038094974364362351691314, −6.15187350174025906605567296537, −1.07858708663533899288725291379,
0.17981795387536254793030726404, 3.40307888029921847556182480108, 5.90287324361262780370803770070, 8.312713518541389264976657579138, 9.665039541253742495829836529068, 10.21364028092421034275657990077, 11.83055979856436492415263029331, 12.05978957709643790293881325655, 15.56111568785296642319313831664, 16.08398214674150673615560942046