| L(s) = 1 | + (−1.23 − 3.80i)2-s + (−12.9 + 9.40i)4-s + (−11.5 − 3.74i)5-s + (−64.3 − 88.6i)7-s + (51.7 + 37.6i)8-s + 48.5i·10-s + (27.8 − 400. i)11-s + (−158. + 51.5i)13-s + (−257. + 354. i)14-s + (79.1 − 243. i)16-s + (−106. + 327. i)17-s + (1.27e3 − 1.75e3i)19-s + (184. − 59.9i)20-s + (−1.55e3 + 388. i)22-s + 2.37e3i·23-s + ⋯ |
| L(s) = 1 | + (−0.218 − 0.672i)2-s + (−0.404 + 0.293i)4-s + (−0.206 − 0.0670i)5-s + (−0.496 − 0.683i)7-s + (0.286 + 0.207i)8-s + 0.153i·10-s + (0.0693 − 0.997i)11-s + (−0.260 + 0.0845i)13-s + (−0.351 + 0.483i)14-s + (0.0772 − 0.237i)16-s + (−0.0893 + 0.274i)17-s + (0.808 − 1.11i)19-s + (0.103 − 0.0335i)20-s + (−0.686 + 0.171i)22-s + 0.935i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0346 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0346 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2078426509\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2078426509\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 + 3.80i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (-27.8 + 400. i)T \) |
| good | 5 | \( 1 + (11.5 + 3.74i)T + (2.52e3 + 1.83e3i)T^{2} \) |
| 7 | \( 1 + (64.3 + 88.6i)T + (-5.19e3 + 1.59e4i)T^{2} \) |
| 13 | \( 1 + (158. - 51.5i)T + (3.00e5 - 2.18e5i)T^{2} \) |
| 17 | \( 1 + (106. - 327. i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.27e3 + 1.75e3i)T + (-7.65e5 - 2.35e6i)T^{2} \) |
| 23 | \( 1 - 2.37e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (6.17e3 - 4.48e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-1.36e3 - 4.20e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (473. - 343. i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (377. + 274. i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 5.42e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (1.49e4 - 2.05e4i)T + (-7.08e7 - 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.17e4 - 7.07e3i)T + (3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-2.46e4 - 3.39e4i)T + (-2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.37e4 - 4.46e3i)T + (6.83e8 + 4.96e8i)T^{2} \) |
| 67 | \( 1 - 3.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (9.56e3 + 3.10e3i)T + (1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-3.62e4 - 4.98e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (5.99e4 - 1.94e4i)T + (2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-611. + 1.88e3i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 - 5.26e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (2.97e4 + 9.16e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−11.59652336139583103928035854588, −10.99253171159809054620216975361, −9.886559805248262951507569330041, −9.058007708141248517651369017953, −7.899759546472437179060562302079, −6.83482973668250034030102457820, −5.37046397295161247347888600722, −3.94355128018624269154486574750, −2.97950983432000070523902614636, −1.20295196087378163537255314587,
0.07439294452158739895357626655, 2.06426847099430301585663009479, 3.76395736134901111536528856489, 5.15684354103290839164004054821, 6.18516344067148581456902807035, 7.30791325267409574388855313641, 8.172901007274701391370971936566, 9.488568169823408320522205069099, 9.933198564341205900980829227119, 11.46588775102385664744112180336
