| L(s) = 1 | + (−0.680 + 0.229i)2-s + (−1.71 + 0.204i)3-s + (−1.18 + 0.898i)4-s + (1.34 − 0.536i)5-s + (1.12 − 0.533i)6-s + (−2.55 − 1.18i)7-s + (1.40 − 2.07i)8-s + (2.91 − 0.702i)9-s + (−0.792 + 0.673i)10-s + (−1.16 − 4.18i)11-s + (1.84 − 1.78i)12-s + (0.962 − 0.510i)13-s + (2.00 + 0.218i)14-s + (−2.20 + 1.19i)15-s + (0.313 − 1.12i)16-s + (−1.77 − 3.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.481 + 0.162i)2-s + (−0.993 + 0.117i)3-s + (−0.590 + 0.449i)4-s + (0.601 − 0.239i)5-s + (0.458 − 0.217i)6-s + (−0.964 − 0.446i)7-s + (0.496 − 0.732i)8-s + (0.972 − 0.234i)9-s + (−0.250 + 0.212i)10-s + (−0.350 − 1.26i)11-s + (0.533 − 0.515i)12-s + (0.267 − 0.141i)13-s + (0.536 + 0.0583i)14-s + (−0.569 + 0.309i)15-s + (0.0783 − 0.282i)16-s + (−0.429 − 0.928i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0168 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0168 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.280025 - 0.275333i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.280025 - 0.275333i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.71 - 0.204i)T \) |
| 59 | \( 1 + (-7.06 - 3.01i)T \) |
| good | 2 | \( 1 + (0.680 - 0.229i)T + (1.59 - 1.21i)T^{2} \) |
| 5 | \( 1 + (-1.34 + 0.536i)T + (3.62 - 3.43i)T^{2} \) |
| 7 | \( 1 + (2.55 + 1.18i)T + (4.53 + 5.33i)T^{2} \) |
| 11 | \( 1 + (1.16 + 4.18i)T + (-9.42 + 5.67i)T^{2} \) |
| 13 | \( 1 + (-0.962 + 0.510i)T + (7.29 - 10.7i)T^{2} \) |
| 17 | \( 1 + (1.77 + 3.82i)T + (-11.0 + 12.9i)T^{2} \) |
| 19 | \( 1 + (-0.143 - 0.0315i)T + (17.2 + 7.97i)T^{2} \) |
| 23 | \( 1 + (1.06 + 6.49i)T + (-21.7 + 7.34i)T^{2} \) |
| 29 | \( 1 + (1.51 - 4.48i)T + (-23.0 - 17.5i)T^{2} \) |
| 31 | \( 1 + (-0.503 - 2.28i)T + (-28.1 + 13.0i)T^{2} \) |
| 37 | \( 1 + (-2.89 + 1.96i)T + (13.6 - 34.3i)T^{2} \) |
| 41 | \( 1 + (-4.23 - 0.694i)T + (38.8 + 13.0i)T^{2} \) |
| 43 | \( 1 + (7.33 + 2.03i)T + (36.8 + 22.1i)T^{2} \) |
| 47 | \( 1 + (-0.818 + 2.05i)T + (-34.1 - 32.3i)T^{2} \) |
| 53 | \( 1 + (5.95 + 5.05i)T + (8.57 + 52.3i)T^{2} \) |
| 61 | \( 1 + (4.01 + 11.9i)T + (-48.5 + 36.9i)T^{2} \) |
| 67 | \( 1 + (-5.30 - 3.59i)T + (24.7 + 62.2i)T^{2} \) |
| 71 | \( 1 + (11.2 + 4.47i)T + (51.5 + 48.8i)T^{2} \) |
| 73 | \( 1 + (1.11 - 10.2i)T + (-71.2 - 15.6i)T^{2} \) |
| 79 | \( 1 + (-15.1 - 9.09i)T + (37.0 + 69.7i)T^{2} \) |
| 83 | \( 1 + (-0.675 + 12.4i)T + (-82.5 - 8.97i)T^{2} \) |
| 89 | \( 1 + (4.75 + 1.60i)T + (70.8 + 53.8i)T^{2} \) |
| 97 | \( 1 + (1.41 + 13.0i)T + (-94.7 + 20.8i)T^{2} \) |
| 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
−12.68246064693358070902113507441, −11.31690125551092193890811909586, −10.30344666747944080949787094986, −9.533319560430411453615340681331, −8.536590123458239766458714951984, −7.12616945929909714684399866621, −6.13702539368598523428262476045, −4.92883791272197991187808088030, −3.49442193631680657948588203137, −0.48035554583857819525801382188,
1.90322850529646046605015151803, 4.33970311663136006547291290459, 5.65764319172154522852585679750, 6.38098173416711303328749617979, 7.79300049715650814628123187394, 9.438054031339117757340133792858, 9.858233096469119271985406166823, 10.72034437430603436592281880587, 11.89111211691657633424740442626, 12.99162538253463660966773737414
