| L(s) = 1 | + (−2.30 + 3.99i)2-s + (4.25 − 7.93i)3-s + (−2.66 − 4.61i)4-s + (−21.0 − 13.4i)5-s + (21.9 + 35.3i)6-s + (−45.6 − 26.3i)7-s − 49.2·8-s + (−44.8 − 67.4i)9-s + (102. − 53.3i)10-s + (−1.90 − 1.09i)11-s + (−47.9 + 1.51i)12-s + (226. − 131. i)13-s + (210. − 121. i)14-s + (−196. + 110. i)15-s + (156. − 270. i)16-s − 487.·17-s + ⋯ |
| L(s) = 1 | + (−0.577 + 0.999i)2-s + (0.472 − 0.881i)3-s + (−0.166 − 0.288i)4-s + (−0.843 − 0.536i)5-s + (0.608 + 0.981i)6-s + (−0.930 − 0.537i)7-s − 0.769·8-s + (−0.553 − 0.832i)9-s + (1.02 − 0.533i)10-s + (−0.0157 − 0.00907i)11-s + (−0.332 + 0.0105i)12-s + (1.34 − 0.775i)13-s + (1.07 − 0.620i)14-s + (−0.871 + 0.490i)15-s + (0.611 − 1.05i)16-s − 1.68·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.153 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.367942 - 0.429596i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.367942 - 0.429596i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.25 + 7.93i)T \) |
| 5 | \( 1 + (21.0 + 13.4i)T \) |
| good | 2 | \( 1 + (2.30 - 3.99i)T + (-8 - 13.8i)T^{2} \) |
| 7 | \( 1 + (45.6 + 26.3i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (1.90 + 1.09i)T + (7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-226. + 131. i)T + (1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + 487.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 8.08T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-178. - 309. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-25.7 - 14.8i)T + (3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (534. + 925. i)T + (-4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.72e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-2.23e3 + 1.29e3i)T + (1.41e6 - 2.44e6i)T^{2} \) |
| 43 | \( 1 + (597. + 345. i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (-817. + 1.41e3i)T + (-2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 3.35e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-1.24e3 + 719. i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (256. - 444. i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-3.80e3 + 2.19e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + 4.43e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.04e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + (2.00e3 - 3.47e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-3.25e3 + 5.64e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + 7.63e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (1.10e4 + 6.35e3i)T + (4.42e7 + 7.66e7i)T^{2} \) |
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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
−15.24937089745388762978707770469, −13.47002756301993492579124402400, −12.74920426434584406470503091895, −11.30825728624301555520784474781, −9.176959391102573428251995324126, −8.303780075070951328850778440453, −7.25577544729643255529267754366, −6.20462155997806131443136328744, −3.48210441362624699433917152589, −0.39392924809034658198704396129,
2.65713162104334214442228062662, 3.93591917645316977043005534154, 6.41979271561097507737343586380, 8.630972101820485004916417158643, 9.343276000296225307338382996504, 10.79605893976929148010141680670, 11.27860849639771241317451167572, 12.80412252173949170714186139078, 14.42126647187064126605734316473, 15.64951748815464175216150870787
