L(s) = 1 | + (0.420 + 0.242i)5-s + 1.92i·7-s + 0.610·11-s + (−0.889 − 1.54i)13-s + (−3.89 − 2.25i)17-s + (−1.30 − 4.15i)19-s + (0.121 + 0.209i)23-s + (−2.38 − 4.12i)25-s + (−5.18 + 2.99i)29-s − 0.423i·31-s + (−0.467 + 0.809i)35-s − 1.98·37-s + (−5.42 − 3.13i)41-s + (3.24 + 1.87i)43-s + (−2.88 − 4.99i)47-s + ⋯ |
L(s) = 1 | + (0.187 + 0.108i)5-s + 0.728i·7-s + 0.184·11-s + (−0.246 − 0.427i)13-s + (−0.945 − 0.546i)17-s + (−0.298 − 0.954i)19-s + (0.0252 + 0.0437i)23-s + (−0.476 − 0.825i)25-s + (−0.962 + 0.555i)29-s − 0.0759i·31-s + (−0.0789 + 0.136i)35-s − 0.326·37-s + (−0.847 − 0.489i)41-s + (0.495 + 0.286i)43-s + (−0.420 − 0.728i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8213986970\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8213986970\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (1.30 + 4.15i)T \) |
good | 5 | \( 1 + (-0.420 - 0.242i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 1.92iT - 7T^{2} \) |
| 11 | \( 1 - 0.610T + 11T^{2} \) |
| 13 | \( 1 + (0.889 + 1.54i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (3.89 + 2.25i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-0.121 - 0.209i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (5.18 - 2.99i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.423iT - 31T^{2} \) |
| 37 | \( 1 + 1.98T + 37T^{2} \) |
| 41 | \( 1 + (5.42 + 3.13i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.24 - 1.87i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.88 + 4.99i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2.85 - 1.64i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.16 + 3.75i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.59 + 2.75i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.13 + 2.38i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7.60 + 13.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.34 + 5.78i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-13.3 - 7.71i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8.99T + 83T^{2} \) |
| 89 | \( 1 + (8.14 - 4.70i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.18 + 5.52i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.707898882031265296286677026120, −7.87574384206409121682676283439, −6.95124476781509233573862235486, −6.36850169011323858771063316888, −5.40940220246293301354440336025, −4.80061900965952104608077518600, −3.72865072080569044764170980533, −2.66920283617399145911914780019, −1.95637634383497754182725113334, −0.25409911176577484264675925209,
1.37972093436697573627364033322, 2.29581116018623798765034466169, 3.68942642333013127388791876488, 4.16556590657552354507635942036, 5.19112852368429403390893288764, 6.06699423299216434327884757976, 6.79845737209437991300511178909, 7.53568518069405465638212808839, 8.297272634083590171304627124870, 9.100448435116157297558426951065