| L(s) = 1 | + (26.2 + 4.15i)2-s + (39.3 + 77.1i)3-s + (428. + 139. i)4-s + (614. + 112. i)5-s + (710. + 2.18e3i)6-s + (−2.10e3 − 2.10e3i)7-s + (4.59e3 + 2.34e3i)8-s + (−548. + 755. i)9-s + (1.56e4 + 5.50e3i)10-s + (−1.89e4 + 1.37e4i)11-s + (6.09e3 + 3.84e4i)12-s + (2.94e4 − 4.67e3i)13-s + (−4.65e4 − 6.40e4i)14-s + (1.54e4 + 5.18e4i)15-s + (1.76e4 + 1.28e4i)16-s + (4.35e4 − 8.54e4i)17-s + ⋯ |
| L(s) = 1 | + (1.64 + 0.259i)2-s + (0.485 + 0.952i)3-s + (1.67 + 0.543i)4-s + (0.983 + 0.179i)5-s + (0.548 + 1.68i)6-s + (−0.877 − 0.877i)7-s + (1.12 + 0.571i)8-s + (−0.0836 + 0.115i)9-s + (1.56 + 0.550i)10-s + (−1.29 + 0.939i)11-s + (0.293 + 1.85i)12-s + (1.03 − 0.163i)13-s + (−1.21 − 1.66i)14-s + (0.306 + 1.02i)15-s + (0.269 + 0.195i)16-s + (0.521 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(4.33584 + 2.42733i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.33584 + 2.42733i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-614. - 112. i)T \) |
| good | 2 | \( 1 + (-26.2 - 4.15i)T + (243. + 79.1i)T^{2} \) |
| 3 | \( 1 + (-39.3 - 77.1i)T + (-3.85e3 + 5.30e3i)T^{2} \) |
| 7 | \( 1 + (2.10e3 + 2.10e3i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + (1.89e4 - 1.37e4i)T + (6.62e7 - 2.03e8i)T^{2} \) |
| 13 | \( 1 + (-2.94e4 + 4.67e3i)T + (7.75e8 - 2.52e8i)T^{2} \) |
| 17 | \( 1 + (-4.35e4 + 8.54e4i)T + (-4.10e9 - 5.64e9i)T^{2} \) |
| 19 | \( 1 + (7.68e4 - 2.49e4i)T + (1.37e10 - 9.98e9i)T^{2} \) |
| 23 | \( 1 + (2.13e3 - 1.34e4i)T + (-7.44e10 - 2.41e10i)T^{2} \) |
| 29 | \( 1 + (7.74e5 + 2.51e5i)T + (4.04e11 + 2.94e11i)T^{2} \) |
| 31 | \( 1 + (-7.44e4 - 2.29e5i)T + (-6.90e11 + 5.01e11i)T^{2} \) |
| 37 | \( 1 + (4.25e5 + 2.68e6i)T + (-3.34e12 + 1.08e12i)T^{2} \) |
| 41 | \( 1 + (-1.39e6 - 1.01e6i)T + (2.46e12 + 7.59e12i)T^{2} \) |
| 43 | \( 1 + (3.27e6 - 3.27e6i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (-3.34e6 + 1.70e6i)T + (1.39e13 - 1.92e13i)T^{2} \) |
| 53 | \( 1 + (-4.92e6 - 9.67e6i)T + (-3.65e13 + 5.03e13i)T^{2} \) |
| 59 | \( 1 + (-3.97e5 + 5.47e5i)T + (-4.53e13 - 1.39e14i)T^{2} \) |
| 61 | \( 1 + (-4.68e6 + 3.40e6i)T + (5.92e13 - 1.82e14i)T^{2} \) |
| 67 | \( 1 + (1.38e7 - 2.71e7i)T + (-2.38e14 - 3.28e14i)T^{2} \) |
| 71 | \( 1 + (2.70e5 - 8.32e5i)T + (-5.22e14 - 3.79e14i)T^{2} \) |
| 73 | \( 1 + (7.08e6 - 4.47e7i)T + (-7.66e14 - 2.49e14i)T^{2} \) |
| 79 | \( 1 + (-1.37e7 - 4.48e6i)T + (1.22e15 + 8.91e14i)T^{2} \) |
| 83 | \( 1 + (1.59e7 + 8.14e6i)T + (1.32e15 + 1.82e15i)T^{2} \) |
| 89 | \( 1 + (-3.38e5 - 4.65e5i)T + (-1.21e15 + 3.74e15i)T^{2} \) |
| 97 | \( 1 + (1.33e8 - 6.82e7i)T + (4.60e15 - 6.34e15i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.63503371042667006887686835934, −14.57429785437055618556231662800, −13.49337806908072215409568350223, −12.80046216645804884672002488223, −10.62649071014850691684608401720, −9.613246761070823304749063416282, −7.04647521954370486877607716268, −5.55030016097387583709045823828, −4.09201816310341536076057742298, −2.82641011609934153210748462235,
1.95762085380004598145902421070, 3.13081492877878051966164569576, 5.54285679761286808447614702231, 6.38361924071057288760386875823, 8.556613182419394079286446872754, 10.64819607347705534019942361375, 12.44333983770477752229207665236, 13.23658061547954432030740388948, 13.62315967061575797807646806641, 15.11277567388345727515968842247
