| L(s) = 1 | + (3.24 − 2.35i)2-s + (−34.5 + 106. i)4-s + (−354. − 257. i)5-s + (116. − 359. i)7-s + (297. + 914. i)8-s − 1.75e3·10-s + (−1.25e3 − 4.23e3i)11-s + (−2.37e3 + 1.72e3i)13-s + (−468. − 1.44e3i)14-s + (−8.46e3 − 6.15e3i)16-s + (1.03e4 + 7.51e3i)17-s + (8.79e3 + 2.70e4i)19-s + (3.96e4 − 2.87e4i)20-s + (−1.40e4 − 1.07e4i)22-s + 7.72e4·23-s + ⋯ |
| L(s) = 1 | + (0.286 − 0.208i)2-s + (−0.270 + 0.831i)4-s + (−1.26 − 0.920i)5-s + (0.128 − 0.395i)7-s + (0.205 + 0.631i)8-s − 0.554·10-s + (−0.284 − 0.958i)11-s + (−0.300 + 0.218i)13-s + (−0.0455 − 0.140i)14-s + (−0.516 − 0.375i)16-s + (0.510 + 0.371i)17-s + (0.294 + 0.905i)19-s + (1.10 − 0.804i)20-s + (−0.281 − 0.215i)22-s + 1.32·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.31596 + 0.391741i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.31596 + 0.391741i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + (1.25e3 + 4.23e3i)T \) |
| good | 2 | \( 1 + (-3.24 + 2.35i)T + (39.5 - 121. i)T^{2} \) |
| 5 | \( 1 + (354. + 257. i)T + (2.41e4 + 7.43e4i)T^{2} \) |
| 7 | \( 1 + (-116. + 359. i)T + (-6.66e5 - 4.84e5i)T^{2} \) |
| 13 | \( 1 + (2.37e3 - 1.72e3i)T + (1.93e7 - 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-1.03e4 - 7.51e3i)T + (1.26e8 + 3.90e8i)T^{2} \) |
| 19 | \( 1 + (-8.79e3 - 2.70e4i)T + (-7.23e8 + 5.25e8i)T^{2} \) |
| 23 | \( 1 - 7.72e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (6.80e4 - 2.09e5i)T + (-1.39e10 - 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-1.34e5 + 9.79e4i)T + (8.50e9 - 2.61e10i)T^{2} \) |
| 37 | \( 1 + (1.13e5 - 3.48e5i)T + (-7.68e10 - 5.57e10i)T^{2} \) |
| 41 | \( 1 + (8.28e4 + 2.55e5i)T + (-1.57e11 + 1.14e11i)T^{2} \) |
| 43 | \( 1 - 8.21e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-9.56e4 - 2.94e5i)T + (-4.09e11 + 2.97e11i)T^{2} \) |
| 53 | \( 1 + (-8.18e5 + 5.94e5i)T + (3.63e11 - 1.11e12i)T^{2} \) |
| 59 | \( 1 + (-2.47e4 + 7.60e4i)T + (-2.01e12 - 1.46e12i)T^{2} \) |
| 61 | \( 1 + (-3.94e5 - 2.86e5i)T + (9.71e11 + 2.98e12i)T^{2} \) |
| 67 | \( 1 - 6.59e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + (1.97e6 + 1.43e6i)T + (2.81e12 + 8.64e12i)T^{2} \) |
| 73 | \( 1 + (1.32e6 - 4.07e6i)T + (-8.93e12 - 6.49e12i)T^{2} \) |
| 79 | \( 1 + (-6.30e6 + 4.58e6i)T + (5.93e12 - 1.82e13i)T^{2} \) |
| 83 | \( 1 + (-8.18e5 - 5.94e5i)T + (8.38e12 + 2.58e13i)T^{2} \) |
| 89 | \( 1 + 5.65e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (-1.15e7 + 8.36e6i)T + (2.49e13 - 7.68e13i)T^{2} \) |
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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
−12.49588912522223998159455452241, −11.79716654473453257355450728729, −10.79491279080214269340686214760, −8.975992074482199241579264114904, −8.170808967775353028507132736202, −7.35543239616863193668029386815, −5.25049432771082231240655347483, −4.13548214465447428592586066998, −3.21195758964392162781371428619, −0.914716774653214296794244937662,
0.54397833507700128106977845532, 2.63062828873093251450401919452, 4.19052833698450736090132729248, 5.31489224530154648428755897254, 6.85585301622238804586035553691, 7.62896691316871072710111727785, 9.224622730062696910737575782285, 10.37019090288485873838189816681, 11.31210389909756908769138000084, 12.33421726263145435805460548094
