L(s) = 1 | + (−0.707 + 0.707i)2-s + (0.707 + 0.707i)3-s − 1.00i·4-s + (0.528 + 2.17i)5-s − 1.00·6-s + (0.904 + 0.904i)7-s + (0.707 + 0.707i)8-s + 1.00i·9-s + (−1.90 − 1.16i)10-s + 2.66·11-s + (0.707 − 0.707i)12-s + (−0.143 − 0.143i)13-s − 1.27·14-s + (−1.16 + 1.90i)15-s − 1.00·16-s + (3.29 + 3.29i)17-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.499i)2-s + (0.408 + 0.408i)3-s − 0.500i·4-s + (0.236 + 0.971i)5-s − 0.408·6-s + (0.341 + 0.341i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s + (−0.603 − 0.367i)10-s + 0.802·11-s + (0.204 − 0.204i)12-s + (−0.0398 − 0.0398i)13-s − 0.341·14-s + (−0.300 + 0.493i)15-s − 0.250·16-s + (0.799 + 0.799i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.405 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.757091 + 1.16418i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.757091 + 1.16418i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.707 - 0.707i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.528 - 2.17i)T \) |
| 19 | \( 1 + (4.05 + 1.59i)T \) |
good | 7 | \( 1 + (-0.904 - 0.904i)T + 7iT^{2} \) |
| 11 | \( 1 - 2.66T + 11T^{2} \) |
| 13 | \( 1 + (0.143 + 0.143i)T + 13iT^{2} \) |
| 17 | \( 1 + (-3.29 - 3.29i)T + 17iT^{2} \) |
| 23 | \( 1 + (-1.75 + 1.75i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.17T + 29T^{2} \) |
| 31 | \( 1 - 2.62iT - 31T^{2} \) |
| 37 | \( 1 + (-0.984 + 0.984i)T - 37iT^{2} \) |
| 41 | \( 1 - 3.74iT - 41T^{2} \) |
| 43 | \( 1 + (2.01 - 2.01i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.24 - 3.24i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.54 - 2.54i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 8.90T + 61T^{2} \) |
| 67 | \( 1 + (-4.50 + 4.50i)T - 67iT^{2} \) |
| 71 | \( 1 + 2.55iT - 71T^{2} \) |
| 73 | \( 1 + (5.25 - 5.25i)T - 73iT^{2} \) |
| 79 | \( 1 + 4.22T + 79T^{2} \) |
| 83 | \( 1 + (-6.55 + 6.55i)T - 83iT^{2} \) |
| 89 | \( 1 + 4.94T + 89T^{2} \) |
| 97 | \( 1 + (-11.8 + 11.8i)T - 97iT^{2} \) |
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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
−10.75885805310586569522569740434, −10.09566354452157162991579318952, −9.193407008456523727440220000118, −8.441180289452606549352041376714, −7.50632920162434316190966378074, −6.55040295831606183827779046406, −5.75351858280540350982983953645, −4.43283512611468636615562963721, −3.19345870023480770509043091277, −1.85229659862933642519376542140,
0.964224966801169211451947229202, 2.05716475673625575243449060031, 3.59292258905153431047240686412, 4.62673914882723067410587927085, 5.89896958380785128882651186955, 7.15917033236392674627124854057, 7.968235678634431574243281352679, 8.835204886167408461315513943407, 9.426075215932657854794471818914, 10.31797480454316602236917033662