| L(s) = 1 | + (−0.0147 − 0.00259i)2-s + (−1.71 + 0.214i)3-s + (−1.87 − 0.683i)4-s + (1.54 + 0.562i)5-s + (0.0258 + 0.00129i)6-s + (2.21 + 1.45i)7-s + (0.0517 + 0.0298i)8-s + (2.90 − 0.738i)9-s + (−0.0212 − 0.0122i)10-s + (1.57 + 4.31i)11-s + (3.37 + 0.771i)12-s + (−1.52 + 4.19i)13-s + (−0.0287 − 0.0271i)14-s + (−2.77 − 0.635i)15-s + (3.06 + 2.57i)16-s + (1.88 − 3.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.0103 − 0.00183i)2-s + (−0.992 + 0.124i)3-s + (−0.939 − 0.341i)4-s + (0.691 + 0.251i)5-s + (0.0105 + 0.000528i)6-s + (0.835 + 0.549i)7-s + (0.0182 + 0.0105i)8-s + (0.969 − 0.246i)9-s + (−0.00672 − 0.00388i)10-s + (0.473 + 1.30i)11-s + (0.974 + 0.222i)12-s + (−0.423 + 1.16i)13-s + (−0.00767 − 0.00724i)14-s + (−0.717 − 0.163i)15-s + (0.765 + 0.642i)16-s + (0.457 − 0.792i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 - 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.782195 + 0.343665i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.782195 + 0.343665i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.71 - 0.214i)T \) |
| 7 | \( 1 + (-2.21 - 1.45i)T \) |
| good | 2 | \( 1 + (0.0147 + 0.00259i)T + (1.87 + 0.684i)T^{2} \) |
| 5 | \( 1 + (-1.54 - 0.562i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-1.57 - 4.31i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.52 - 4.19i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.88 + 3.26i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.27 - 0.734i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.31 + 1.29i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.83 + 7.78i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (2.87 - 7.89i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + 0.794T + 37T^{2} \) |
| 41 | \( 1 + (-3.97 - 1.44i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.303 - 1.71i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (1.54 - 0.563i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-3.66 + 2.11i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (8.85 - 7.43i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.02 - 2.82i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (1.88 + 10.6i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.66 + 3.84i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 15.8iT - 73T^{2} \) |
| 79 | \( 1 + (-0.276 + 1.56i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-4.61 + 1.68i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (3.59 + 6.22i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.56 + 0.276i)T + (91.1 + 33.1i)T^{2} \) |
| 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
−12.50618385639028346110073189495, −11.80375885364851563507174040977, −10.67599646386209234463182225177, −9.644123091754747830726317209263, −9.137578307911491991952292189135, −7.39834517599988922103747874914, −6.23584946417730721084838960245, −5.05871406751006272387248271428, −4.45751551136903536586503161006, −1.74957699571190356270650366559,
1.02950615306312294366649536904, 3.74117921405606810904841092802, 5.14807762884543057845024307232, 5.72824011816990289892866055871, 7.35254031404401753627330779365, 8.395995499783669539746272202559, 9.506696527374516280808394485602, 10.62401192779098323974216585364, 11.32594998662234143358027495866, 12.71176765906388436978353314792
