L(s) = 1 | + (0.5 − 0.866i)2-s + (−2.35 − 0.630i)3-s + (−0.499 − 0.866i)4-s + (−0.0214 − 2.23i)5-s + (−1.72 + 1.72i)6-s + (−1.71 − 0.460i)7-s − 0.999·8-s + (2.54 + 1.46i)9-s + (−1.94 − 1.09i)10-s + 2.57i·11-s + (0.630 + 2.35i)12-s + (−0.427 − 0.739i)13-s + (−1.25 + 1.25i)14-s + (−1.35 + 5.27i)15-s + (−0.5 + 0.866i)16-s + (−0.757 − 0.437i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−1.35 − 0.364i)3-s + (−0.249 − 0.433i)4-s + (−0.00960 − 0.999i)5-s + (−0.703 + 0.703i)6-s + (−0.649 − 0.173i)7-s − 0.353·8-s + (0.847 + 0.489i)9-s + (−0.615 − 0.347i)10-s + 0.776i·11-s + (0.182 + 0.679i)12-s + (−0.118 − 0.205i)13-s + (−0.336 + 0.336i)14-s + (−0.350 + 1.36i)15-s + (−0.125 + 0.216i)16-s + (−0.183 − 0.106i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.585 - 0.810i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.134556 + 0.263076i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.134556 + 0.263076i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.0214 + 2.23i)T \) |
| 37 | \( 1 + (-1.23 + 5.95i)T \) |
good | 3 | \( 1 + (2.35 + 0.630i)T + (2.59 + 1.5i)T^{2} \) |
| 7 | \( 1 + (1.71 + 0.460i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 - 2.57iT - 11T^{2} \) |
| 13 | \( 1 + (0.427 + 0.739i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.757 + 0.437i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.00 - 3.76i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + 3.91T + 23T^{2} \) |
| 29 | \( 1 + (1.14 - 1.14i)T - 29iT^{2} \) |
| 31 | \( 1 + (1.60 + 1.60i)T + 31iT^{2} \) |
| 41 | \( 1 + (-3.54 + 2.04i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + (-6.26 + 6.26i)T - 47iT^{2} \) |
| 53 | \( 1 + (9.72 - 2.60i)T + (45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (3.20 - 0.859i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-2.49 + 9.30i)T + (-52.8 - 30.5i)T^{2} \) |
| 67 | \( 1 + (-0.793 + 2.95i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.32 - 2.29i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-8.92 + 8.92i)T - 73iT^{2} \) |
| 79 | \( 1 + (-1.90 + 7.09i)T + (-68.4 - 39.5i)T^{2} \) |
| 83 | \( 1 + (8.30 - 2.22i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-4.21 - 15.7i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 - 2.34iT - 97T^{2} \) |
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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
−10.97799612519703405157765482757, −10.07218927532509393239723418363, −9.300541055948564314671625351350, −7.914439227735721937593499515504, −6.65232853230979327719867232238, −5.74371754505772476076786947000, −4.93365424221084373063731911133, −3.84655908081512845237772376221, −1.79133186365126244954148827996, −0.20374894836351664066642204302,
2.95861069542295452733601320068, 4.24074329057753150154870715870, 5.44345460343513404285377379018, 6.30927600057393299498207682101, 6.74085280846900573036532582970, 8.057009318293892471375060514526, 9.409075415136095624863460668647, 10.34366193494798661394958568752, 11.20454939532783874488621730218, 11.76590937202513036981487203648