L(s) = 1 | + (−0.965 + 0.258i)2-s + (0.258 + 0.965i)3-s + (0.866 − 0.499i)4-s + (−0.991 + 2.00i)5-s + (−0.499 − 0.866i)6-s + (1.93 − 1.93i)7-s + (−0.707 + 0.707i)8-s + (−0.866 + 0.499i)9-s + (0.439 − 2.19i)10-s − 6.13·11-s + (0.707 + 0.707i)12-s + (−6.20 − 1.66i)13-s + (−1.36 + 2.37i)14-s + (−2.19 − 0.439i)15-s + (0.500 − 0.866i)16-s + (0.956 + 3.56i)17-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.183i)2-s + (0.149 + 0.557i)3-s + (0.433 − 0.249i)4-s + (−0.443 + 0.896i)5-s + (−0.204 − 0.353i)6-s + (0.731 − 0.731i)7-s + (−0.249 + 0.249i)8-s + (−0.288 + 0.166i)9-s + (0.138 − 0.693i)10-s − 1.85·11-s + (0.204 + 0.204i)12-s + (−1.72 − 0.461i)13-s + (−0.365 + 0.633i)14-s + (−0.566 − 0.113i)15-s + (0.125 − 0.216i)16-s + (0.231 + 0.865i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.931 + 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.931 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0442690 - 0.235860i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0442690 - 0.235860i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 - 0.258i)T \) |
| 3 | \( 1 + (-0.258 - 0.965i)T \) |
| 5 | \( 1 + (0.991 - 2.00i)T \) |
| 19 | \( 1 + (-3.01 - 3.15i)T \) |
good | 7 | \( 1 + (-1.93 + 1.93i)T - 7iT^{2} \) |
| 11 | \( 1 + 6.13T + 11T^{2} \) |
| 13 | \( 1 + (6.20 + 1.66i)T + (11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (-0.956 - 3.56i)T + (-14.7 + 8.5i)T^{2} \) |
| 23 | \( 1 + (-0.0627 + 0.234i)T + (-19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (3.45 + 5.99i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 7.31iT - 31T^{2} \) |
| 37 | \( 1 + (7.48 + 7.48i)T + 37iT^{2} \) |
| 41 | \( 1 + (-4.33 - 2.50i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (5.48 - 1.46i)T + (37.2 - 21.5i)T^{2} \) |
| 47 | \( 1 + (7.64 + 2.04i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-1.48 - 0.398i)T + (45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (0.890 - 1.54i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.98 - 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.96 + 7.33i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (2.59 + 1.50i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.59 - 0.428i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (2.06 - 3.56i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.54 + 4.54i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.75 - 6.50i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.39 + 0.640i)T + (84.0 - 48.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.78217360273491166093987204317, −10.32724398927686877880265686619, −9.848348614334720065720882796104, −8.297263992353082329903773734127, −7.69326214200257928561791230251, −7.24474676515032186709446387644, −5.66820045828386905056585617083, −4.78481289908327204396533133622, −3.39134901591863966994260089283, −2.28054772309998475535509568547,
0.15256707923004820180051214136, 1.95830354650855942999272853763, 2.91261374289413035991660782279, 4.98866950515463908234781331270, 5.26040681825110122587479042282, 7.14525948469618558082793273337, 7.69396082474409664962093687076, 8.401116125006689476307286345603, 9.279410397539161444074627304880, 10.05005183301692861237487475877