L(s) = 1 | + (−1.62 − 1.62i)2-s + 3.25i·4-s + (−1.83 − 1.28i)5-s + (−0.707 + 0.707i)7-s + (2.02 − 2.02i)8-s + (0.894 + 5.04i)10-s + 3.03i·11-s + (−2.54 − 2.54i)13-s + 2.29·14-s − 0.0700·16-s + (2.70 + 2.70i)17-s + 6.63i·19-s + (4.16 − 5.96i)20-s + (4.91 − 4.91i)22-s + (−2.99 + 2.99i)23-s + ⋯ |
L(s) = 1 | + (−1.14 − 1.14i)2-s + 1.62i·4-s + (−0.819 − 0.572i)5-s + (−0.267 + 0.267i)7-s + (0.717 − 0.717i)8-s + (0.282 + 1.59i)10-s + 0.914i·11-s + (−0.704 − 0.704i)13-s + 0.612·14-s − 0.0175·16-s + (0.655 + 0.655i)17-s + 1.52i·19-s + (0.931 − 1.33i)20-s + (1.04 − 1.04i)22-s + (−0.623 + 0.623i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.330727 + 0.0982003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.330727 + 0.0982003i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.83 + 1.28i)T \) |
| 7 | \( 1 + (0.707 - 0.707i)T \) |
good | 2 | \( 1 + (1.62 + 1.62i)T + 2iT^{2} \) |
| 11 | \( 1 - 3.03iT - 11T^{2} \) |
| 13 | \( 1 + (2.54 + 2.54i)T + 13iT^{2} \) |
| 17 | \( 1 + (-2.70 - 2.70i)T + 17iT^{2} \) |
| 19 | \( 1 - 6.63iT - 19T^{2} \) |
| 23 | \( 1 + (2.99 - 2.99i)T - 23iT^{2} \) |
| 29 | \( 1 - 5.10T + 29T^{2} \) |
| 31 | \( 1 + 3.28T + 31T^{2} \) |
| 37 | \( 1 + (-6.68 + 6.68i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.36iT - 41T^{2} \) |
| 43 | \( 1 + (-1.74 - 1.74i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.173 - 0.173i)T + 47iT^{2} \) |
| 53 | \( 1 + (8.45 - 8.45i)T - 53iT^{2} \) |
| 59 | \( 1 + 4.60T + 59T^{2} \) |
| 61 | \( 1 + 10.5T + 61T^{2} \) |
| 67 | \( 1 + (8.00 - 8.00i)T - 67iT^{2} \) |
| 71 | \( 1 + 5.47iT - 71T^{2} \) |
| 73 | \( 1 + (5.58 + 5.58i)T + 73iT^{2} \) |
| 79 | \( 1 - 16.1iT - 79T^{2} \) |
| 83 | \( 1 + (-0.998 + 0.998i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.00T + 89T^{2} \) |
| 97 | \( 1 + (8.52 - 8.52i)T - 97iT^{2} \) |
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
−11.78226493979865345451984868776, −10.65153010923698800051461041239, −9.895338391627973218411965085766, −9.197172327874433752247338070176, −7.937869031906812190628700345459, −7.71534199280287124635451334139, −5.77030355930412003375270228756, −4.23716453270412341984524680776, −3.04210148246363666750436109781, −1.49801171794525718295267455815,
0.37742491945095649180632135366, 3.04516889736231943912602408026, 4.67820246973378988042803529494, 6.18248735422642151666625448303, 6.97100513520335568492715620795, 7.66791827838698204480082438214, 8.598941656644384274421388455870, 9.465506913704390596730208301599, 10.41200277858363145776239443312, 11.32583188427847228644650248477