| L(s) = 1 | + (−1.98 − 0.174i)2-s + (1.95 + 0.344i)4-s + (−0.840 + 2.07i)5-s + (1.15 + 0.808i)7-s + (0.0258 + 0.00692i)8-s + (2.03 − 3.97i)10-s + (1.21 − 3.35i)11-s + (0.265 + 3.03i)13-s + (−2.15 − 1.80i)14-s + (−3.78 − 1.37i)16-s + (−3.91 + 1.05i)17-s + (−1.42 + 0.821i)19-s + (−2.35 + 3.76i)20-s + (−3.00 + 6.45i)22-s + (3.67 + 5.24i)23-s + ⋯ |
| L(s) = 1 | + (−1.40 − 0.123i)2-s + (0.978 + 0.172i)4-s + (−0.376 + 0.926i)5-s + (0.436 + 0.305i)7-s + (0.00914 + 0.00244i)8-s + (0.642 − 1.25i)10-s + (0.367 − 1.01i)11-s + (0.0737 + 0.842i)13-s + (−0.576 − 0.483i)14-s + (−0.945 − 0.344i)16-s + (−0.950 + 0.254i)17-s + (−0.326 + 0.188i)19-s + (−0.527 + 0.841i)20-s + (−0.641 + 1.37i)22-s + (0.765 + 1.09i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.213 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.312837 + 0.388413i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.312837 + 0.388413i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.840 - 2.07i)T \) |
| good | 2 | \( 1 + (1.98 + 0.174i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-1.15 - 0.808i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.21 + 3.35i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.265 - 3.03i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (3.91 - 1.05i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (1.42 - 0.821i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.67 - 5.24i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-2.12 + 1.78i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.13 - 6.43i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.751 - 2.80i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (5.92 - 7.06i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.41 - 5.17i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (4.56 - 6.52i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (8.37 - 8.37i)T - 53iT^{2} \) |
| 59 | \( 1 + (-13.4 + 4.89i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.11 - 6.32i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (11.6 - 1.01i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (0.966 + 0.557i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.19 + 8.19i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (3.51 + 4.18i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.529 - 6.05i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-0.0742 - 0.128i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.6 + 5.45i)T + (62.3 - 74.3i)T^{2} \) |
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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
−11.29823325098816548802518915930, −10.64556750566002493502778490805, −9.599104667530696931972052762489, −8.743813929404070663641513606919, −8.096325570081498140315859247876, −7.03838823916450741844675708087, −6.25874561203915649540417388939, −4.56342392704262127059344180467, −3.08710580731479194471910021264, −1.61343878441558886495023409135,
0.53542751125153900756630352204, 2.00172118187418235699160873386, 4.17566641337553446147819584354, 5.04102974860516291121768808860, 6.73742410202722721357726035707, 7.53396456683559541568403424431, 8.442210210892784861307891464365, 8.985332805518381641657529413106, 9.933933691182233739126639937760, 10.76721978367913800324480571232
