| 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
−16.08398214674150673615560942046, −15.56111568785296642319313831664, −12.05978957709643790293881325655, −11.83055979856436492415263029331, −10.21364028092421034275657990077, −9.665039541253742495829836529068, −8.312713518541389264976657579138, −5.90287324361262780370803770070, −3.40307888029921847556182480108, −0.17981795387536254793030726404,
1.07858708663533899288725291379, 6.15187350174025906605567296537, 7.10089038094974364362351691314, 7.80390493755870342174609605998, 9.910979188133131819127205203362, 11.01451770952650539098502130905, 12.35091329376019436484406631548, 14.02681003364583300873090509424, 15.98946223884902685216143022756, 16.72265636985233146775901774119
