| L(s) = 1 | + (0.0590 + 1.41i)2-s + (0.338 − 0.0847i)3-s + (−1.99 + 0.166i)4-s + (−3.32 − 1.57i)5-s + (0.139 + 0.472i)6-s + (1.24 + 4.09i)7-s + (−0.353 − 2.80i)8-s + (−2.53 + 1.35i)9-s + (2.02 − 4.78i)10-s + (−3.21 + 0.477i)11-s + (−0.660 + 0.225i)12-s + (0.312 − 0.872i)13-s + (−5.71 + 1.99i)14-s + (−1.25 − 0.249i)15-s + (3.94 − 0.664i)16-s + (−4.47 + 0.890i)17-s + ⋯ |
| L(s) = 1 | + (0.0417 + 0.999i)2-s + (0.195 − 0.0489i)3-s + (−0.996 + 0.0833i)4-s + (−1.48 − 0.702i)5-s + (0.0570 + 0.193i)6-s + (0.469 + 1.54i)7-s + (−0.124 − 0.992i)8-s + (−0.846 + 0.452i)9-s + (0.639 − 1.51i)10-s + (−0.970 + 0.144i)11-s + (−0.190 + 0.0650i)12-s + (0.0866 − 0.242i)13-s + (−1.52 + 0.533i)14-s + (−0.324 − 0.0645i)15-s + (0.986 − 0.166i)16-s + (−1.08 + 0.215i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0728782 - 0.402225i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0728782 - 0.402225i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0590 - 1.41i)T \) |
| good | 3 | \( 1 + (-0.338 + 0.0847i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (3.32 + 1.57i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (-1.24 - 4.09i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (3.21 - 0.477i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (-0.312 + 0.872i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (4.47 - 0.890i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (3.38 + 3.07i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-2.67 + 0.263i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (-5.76 - 7.77i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-7.52 - 3.11i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (-0.0674 + 1.37i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (3.73 - 4.55i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (0.275 - 1.09i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (9.18 + 6.13i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (0.0408 - 0.0551i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (0.326 + 0.911i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (5.27 - 8.80i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (-4.46 - 2.67i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (-3.78 + 7.07i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-2.17 + 7.16i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-1.55 - 2.32i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-0.228 - 4.64i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (3.53 + 0.348i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (-0.273 + 0.660i)T + (-68.5 - 68.5i)T^{2} \) |
| 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
−12.60523977479011439831784689120, −11.80177574733492773459657669565, −10.78055726867351008345894052796, −8.941498930607154449176058494862, −8.456276948143415278660029649280, −8.039597210602254939220155628036, −6.63783373728912468011028414064, −5.16405056649960848134550044015, −4.73281832122908799341141569912, −2.91524658409085027965051934222,
0.30348677537715696216463289731, 2.78065976994689999282116178116, 3.88980820895477154360923198178, 4.60282395244105669611062982315, 6.60087362823412065895359359628, 7.966371402192048201064214907764, 8.313561307853198892916235721907, 9.940547597766690323938162142304, 10.88441320073135276146123123604, 11.25065257799753057285891687607
