| L(s) = 1 | + (−2.42 + 3.33i)2-s + (−3.56 + 1.15i)3-s + (−2.77 − 8.53i)4-s + (−8.01 − 7.79i)5-s + (4.77 − 14.6i)6-s + 26.4i·7-s + (3.82 + 1.24i)8-s + (−10.4 + 7.61i)9-s + (45.3 − 7.83i)10-s + (15.6 + 11.3i)11-s + (19.7 + 27.2i)12-s + (26.2 + 36.0i)13-s + (−88.2 − 64.1i)14-s + (37.5 + 18.5i)15-s + (44.6 − 32.4i)16-s + (−70.5 − 22.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.856 + 1.17i)2-s + (−0.685 + 0.222i)3-s + (−0.346 − 1.06i)4-s + (−0.716 − 0.697i)5-s + (0.324 − 0.998i)6-s + 1.42i·7-s + (0.169 + 0.0549i)8-s + (−0.388 + 0.282i)9-s + (1.43 − 0.247i)10-s + (0.429 + 0.312i)11-s + (0.475 + 0.654i)12-s + (0.559 + 0.769i)13-s + (−1.68 − 1.22i)14-s + (0.646 + 0.318i)15-s + (0.698 − 0.507i)16-s + (−1.00 − 0.327i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0385579 - 0.355478i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0385579 - 0.355478i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (8.01 + 7.79i)T \) |
| good | 2 | \( 1 + (2.42 - 3.33i)T + (-2.47 - 7.60i)T^{2} \) |
| 3 | \( 1 + (3.56 - 1.15i)T + (21.8 - 15.8i)T^{2} \) |
| 7 | \( 1 - 26.4iT - 343T^{2} \) |
| 11 | \( 1 + (-15.6 - 11.3i)T + (411. + 1.26e3i)T^{2} \) |
| 13 | \( 1 + (-26.2 - 36.0i)T + (-678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (70.5 + 22.9i)T + (3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (6.14 - 18.9i)T + (-5.54e3 - 4.03e3i)T^{2} \) |
| 23 | \( 1 + (-36.2 + 49.8i)T + (-3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-88.7 - 273. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (-25.1 + 77.3i)T + (-2.41e4 - 1.75e4i)T^{2} \) |
| 37 | \( 1 + (230. + 317. i)T + (-1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (176. - 128. i)T + (2.12e4 - 6.55e4i)T^{2} \) |
| 43 | \( 1 - 430. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (174. - 56.7i)T + (8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (59.1 - 19.2i)T + (1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-299. + 217. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (-180. - 130. i)T + (7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + (-423. - 137. i)T + (2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (-126. - 389. i)T + (-2.89e5 + 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-196. + 270. i)T + (-1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-77.3 - 237. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-1.00e3 - 327. i)T + (4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (425. + 309. i)T + (2.17e5 + 6.70e5i)T^{2} \) |
| 97 | \( 1 + (-469. + 152. i)T + (7.38e5 - 5.36e5i)T^{2} \) |
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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
−17.52276896848375443939334030271, −16.32841397536418835750945263621, −15.88588207340222364821785421504, −14.65788959545483980368852849824, −12.40758974322527213946765295415, −11.33504435817422580374894431345, −9.134013941812705421244614846343, −8.439303353207360786342620909350, −6.56814450249389385695383999466, −5.13178365832079277564432707403,
0.51874482156235786339781433214, 3.56185614144795796023463182847, 6.65071338507794810520495260567, 8.387124882294269167917073590259, 10.31708433989361221791805076299, 11.06776661665527730229089500488, 11.92485016984156864444598754495, 13.59124117858192704160679204580, 15.33591229594351219226827778967, 17.14340428205137527434858615065
