| L(s) = 1 | + (0.0808 − 0.0106i)2-s + (0.442 + 0.896i)3-s + (−1.92 + 0.515i)4-s + (0.199 − 0.174i)5-s + (0.0453 + 0.0678i)6-s + (−2.58 − 0.554i)7-s + (−0.300 + 0.124i)8-s + (−0.608 + 0.793i)9-s + (0.0142 − 0.0162i)10-s + (−6.03 + 0.395i)11-s + (−1.31 − 1.49i)12-s + (−1.97 − 1.97i)13-s + (−0.215 − 0.0172i)14-s + (0.245 + 0.101i)15-s + (3.42 − 1.98i)16-s + (3.81 + 1.55i)17-s + ⋯ |
| L(s) = 1 | + (0.0571 − 0.00752i)2-s + (0.255 + 0.517i)3-s + (−0.962 + 0.257i)4-s + (0.0891 − 0.0781i)5-s + (0.0184 + 0.0276i)6-s + (−0.977 − 0.209i)7-s + (−0.106 + 0.0440i)8-s + (−0.202 + 0.264i)9-s + (0.00450 − 0.00514i)10-s + (−1.81 + 0.119i)11-s + (−0.379 − 0.432i)12-s + (−0.547 − 0.547i)13-s + (−0.0574 − 0.00461i)14-s + (0.0632 + 0.0262i)15-s + (0.857 − 0.495i)16-s + (0.926 + 0.376i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0121955 - 0.174259i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0121955 - 0.174259i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.442 - 0.896i)T \) |
| 7 | \( 1 + (2.58 + 0.554i)T \) |
| 17 | \( 1 + (-3.81 - 1.55i)T \) |
| good | 2 | \( 1 + (-0.0808 + 0.0106i)T + (1.93 - 0.517i)T^{2} \) |
| 5 | \( 1 + (-0.199 + 0.174i)T + (0.652 - 4.95i)T^{2} \) |
| 11 | \( 1 + (6.03 - 0.395i)T + (10.9 - 1.43i)T^{2} \) |
| 13 | \( 1 + (1.97 + 1.97i)T + 13iT^{2} \) |
| 19 | \( 1 + (-0.124 - 0.943i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (5.23 + 2.58i)T + (14.0 + 18.2i)T^{2} \) |
| 29 | \( 1 + (0.808 + 4.06i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 + (6.53 - 3.22i)T + (18.8 - 24.5i)T^{2} \) |
| 37 | \( 1 + (0.141 - 2.16i)T + (-36.6 - 4.82i)T^{2} \) |
| 41 | \( 1 + (0.435 - 2.19i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-4.25 - 10.2i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + (-1.94 + 7.25i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.17 + 1.53i)T + (-13.7 + 51.1i)T^{2} \) |
| 59 | \( 1 + (10.4 + 1.37i)T + (56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (-2.64 - 7.78i)T + (-48.3 + 37.1i)T^{2} \) |
| 67 | \( 1 + (-2.89 - 1.67i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.95 - 3.31i)T + (27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-3.65 - 1.24i)T + (57.9 + 44.4i)T^{2} \) |
| 79 | \( 1 + (-7.03 + 14.2i)T + (-48.0 - 62.6i)T^{2} \) |
| 83 | \( 1 + (-1.65 + 3.98i)T + (-58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (1.34 + 0.361i)T + (77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (14.5 - 2.89i)T + (89.6 - 37.1i)T^{2} \) |
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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
−12.24445372082496032668541313554, −10.60470159766498781701498514512, −10.01740005687417429403932874731, −9.385417096954071364845837351229, −8.138414291369994996349662761453, −7.59325829285405432117082834546, −5.80474355689114893062952419639, −5.04788925287027850131806559876, −3.78729225339925751646431417285, −2.82542338835172741213081271505,
0.10900903058998658621620641539, 2.42911959041092733543775235492, 3.69186013476904055465777594556, 5.19614105694680673109270951122, 5.93451495775572580556076371039, 7.32109366488140661305101676588, 8.115599684678223190815873101648, 9.263049232152314234574022663607, 9.869357113939168449094645343407, 10.78368411602991671182188130092
