| L(s) = 1 | + (−14.0 + 3.77i)2-s + (−17.8 + 30.8i)3-s + (128. − 74.0i)4-s + (−91.2 − 91.2i)5-s + (134. − 500. i)6-s + (284. + 76.2i)7-s + (−867. + 867. i)8-s + (−269. − 466. i)9-s + (1.62e3 + 939. i)10-s + (−518. − 1.93e3i)11-s + 5.27e3i·12-s + (−2.11e3 − 597. i)13-s − 4.29e3·14-s + (4.43e3 − 1.18e3i)15-s + (4.18e3 − 7.25e3i)16-s + (1.54e3 − 892. i)17-s + ⋯ |
| L(s) = 1 | + (−1.75 + 0.471i)2-s + (−0.659 + 1.14i)3-s + (2.00 − 1.15i)4-s + (−0.729 − 0.729i)5-s + (0.621 − 2.31i)6-s + (0.829 + 0.222i)7-s + (−1.69 + 1.69i)8-s + (−0.369 − 0.639i)9-s + (1.62 + 0.939i)10-s + (−0.389 − 1.45i)11-s + 3.05i·12-s + (−0.962 − 0.272i)13-s − 1.56·14-s + (1.31 − 0.352i)15-s + (1.02 − 1.77i)16-s + (0.314 − 0.181i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.157403 - 0.119275i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.157403 - 0.119275i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.11e3 + 597. i)T \) |
| good | 2 | \( 1 + (14.0 - 3.77i)T + (55.4 - 32i)T^{2} \) |
| 3 | \( 1 + (17.8 - 30.8i)T + (-364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (91.2 + 91.2i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + (-284. - 76.2i)T + (1.01e5 + 5.88e4i)T^{2} \) |
| 11 | \( 1 + (518. + 1.93e3i)T + (-1.53e6 + 8.85e5i)T^{2} \) |
| 17 | \( 1 + (-1.54e3 + 892. i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-1.47e3 + 5.52e3i)T + (-4.07e7 - 2.35e7i)T^{2} \) |
| 23 | \( 1 + (2.62e3 + 1.51e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (1.40e4 - 2.43e4i)T + (-2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.59e4 + 1.59e4i)T + 8.87e8iT^{2} \) |
| 37 | \( 1 + (8.29e3 + 3.09e4i)T + (-2.22e9 + 1.28e9i)T^{2} \) |
| 41 | \( 1 + (2.08e4 - 5.59e3i)T + (4.11e9 - 2.37e9i)T^{2} \) |
| 43 | \( 1 + (1.13e4 - 6.56e3i)T + (3.16e9 - 5.47e9i)T^{2} \) |
| 47 | \( 1 + (2.23e4 - 2.23e4i)T - 1.07e10iT^{2} \) |
| 53 | \( 1 - 1.35e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + (-4.16e4 - 1.11e4i)T + (3.65e10 + 2.10e10i)T^{2} \) |
| 61 | \( 1 + (7.42e4 + 1.28e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (4.40e5 - 1.17e5i)T + (7.83e10 - 4.52e10i)T^{2} \) |
| 71 | \( 1 + (-1.11e5 + 4.15e5i)T + (-1.10e11 - 6.40e10i)T^{2} \) |
| 73 | \( 1 + (1.60e5 - 1.60e5i)T - 1.51e11iT^{2} \) |
| 79 | \( 1 - 8.12e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + (4.75e5 + 4.75e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 + (-3.39e5 - 1.26e6i)T + (-4.30e11 + 2.48e11i)T^{2} \) |
| 97 | \( 1 + (3.99e5 - 1.49e6i)T + (-7.21e11 - 4.16e11i)T^{2} \) |
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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
−17.99952701009496928500259966802, −16.69807656383415407849181715190, −16.18391368940400007414221632603, −15.06065423238070438366904960417, −11.58808661348578221180783047645, −10.64494247289203155259518754947, −9.101896384367885830188515229864, −7.87026100487398750679756473298, −5.28722171971913805953233201851, −0.28226696015413080481852561195,
1.80247428155898797350088311543, 7.17375365149857314156959018530, 7.73701341650255363077686002122, 10.08046598589475701916899245912, 11.48919867771552487109253642372, 12.28433135895823707704360971064, 15.05169440976899863524519413475, 16.95732409846041047674797735941, 17.85399554397157208798307544259, 18.59364731328929378379795689606
