L(s) = 1 | + (0.5 − 0.866i)3-s + (0.5 + 0.866i)5-s + (−0.499 − 0.866i)9-s + (−1.82 + 3.15i)11-s − 2.64·13-s + 0.999·15-s + (1.82 − 3.15i)17-s + (−1.14 − 1.98i)19-s + (1.82 + 3.15i)23-s + (−0.499 + 0.866i)25-s − 0.999·27-s + 2.35·29-s + (3.14 − 5.44i)31-s + (1.82 + 3.15i)33-s + (2.32 + 4.02i)37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (0.223 + 0.387i)5-s + (−0.166 − 0.288i)9-s + (−0.549 + 0.951i)11-s − 0.733·13-s + 0.258·15-s + (0.442 − 0.765i)17-s + (−0.262 − 0.455i)19-s + (0.380 + 0.658i)23-s + (−0.0999 + 0.173i)25-s − 0.192·27-s + 0.437·29-s + (0.564 − 0.978i)31-s + (0.317 + 0.549i)33-s + (0.381 + 0.661i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.960172046\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.960172046\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (1.82 - 3.15i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 + (-1.82 + 3.15i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.14 + 1.98i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.82 - 3.15i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 + (-3.14 + 5.44i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.32 - 4.02i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 - 9.93T + 43T^{2} \) |
| 47 | \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.64 + 6.31i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.46 - 4.27i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.32 + 4.02i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.35T + 71T^{2} \) |
| 73 | \( 1 + (6.61 - 11.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.14 - 7.18i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.93T + 83T^{2} \) |
| 89 | \( 1 + (-6.11 - 10.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
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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
−8.790047097584187014291935336448, −7.63263218118366294211065548646, −7.50126058948939800398896600150, −6.64163651215406217097928305863, −5.75248944805588621255045944045, −4.91476910805713739448337287771, −4.05643603107130952931830118010, −2.66696997131315755703926017989, −2.43700220885002287821600739176, −0.941404234055516000169391459426,
0.76163827925805889287953545926, 2.22620771641580654064672179646, 3.05806467347491425629041466476, 4.04036076174617339072124236578, 4.82314435419983571841355968577, 5.65691514907184834118000207434, 6.25668516130070046824792996135, 7.44019405649253621651411604907, 8.090530050310651099924767893180, 8.781580625864480940964253590255