L(s) = 1 | + (1.22 − 0.707i)2-s + (−0.866 + 1.5i)3-s + (0.999 − 1.73i)4-s + 2.44i·6-s + (−0.264 − 6.99i)7-s − 2.82i·8-s + (−1.5 − 2.59i)9-s + (2.03 − 3.52i)11-s + (1.73 + 2.99i)12-s − 11.1·13-s + (−5.26 − 8.38i)14-s + (−2.00 − 3.46i)16-s + (−1.44 + 2.50i)17-s + (−3.67 − 2.12i)18-s + (−17.9 + 10.3i)19-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.288 + 0.5i)3-s + (0.249 − 0.433i)4-s + 0.408i·6-s + (−0.0377 − 0.999i)7-s − 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.184 − 0.320i)11-s + (0.144 + 0.249i)12-s − 0.855·13-s + (−0.376 − 0.598i)14-s + (−0.125 − 0.216i)16-s + (−0.0851 + 0.147i)17-s + (−0.204 − 0.117i)18-s + (−0.944 + 0.545i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2778685823\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2778685823\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 + 0.707i)T \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.264 + 6.99i)T \) |
good | 11 | \( 1 + (-2.03 + 3.52i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 11.1T + 169T^{2} \) |
| 17 | \( 1 + (1.44 - 2.50i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (17.9 - 10.3i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (26.5 - 15.3i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 5.75T + 841T^{2} \) |
| 31 | \( 1 + (-3.32 - 1.91i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-7.72 + 4.46i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 56.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 7.16iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (23.8 + 41.2i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (28.8 + 16.6i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-38.3 - 22.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (60.3 - 34.8i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (79.7 + 46.0i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 6.64T + 5.04e3T^{2} \) |
| 73 | \( 1 + (26.2 - 45.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (26.7 + 46.2i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 116.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-85.6 + 49.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + 132.T + 9.40e3T^{2} \) |
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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
−9.708741867000045018737789507136, −8.440210898558269656112003487466, −7.49901226542946273685824411784, −6.51763507989770379918663818316, −5.75447929372817618671792381919, −4.62544225497145577169098014539, −4.06165959190501337151226982135, −3.06607522958817896772994040992, −1.63130975182291730288796342536, −0.06611851706266780426154909773,
1.99595612464210227782480234922, 2.76910504168293560742258362204, 4.26018849087006946682810170097, 5.06562862492268396659982453443, 6.00354641214499374887543813098, 6.62660913989626166919269681659, 7.53740791734257908657557086475, 8.383099703013130752740378882293, 9.196300263749415521706973884720, 10.21455235087913133162606466294