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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.84135 + 0.377404i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.84135 + 0.377404i\) |
\(L(\frac{3}{2})\) |
|
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
−10.89268817071902795863591689455, −9.729809812859201681373016325997, −9.119489559078163381143602138960, −8.050270677037142114470705101644, −7.14697688274713886819664976560, −5.98669092662067251657664770152, −5.18310145283099662955469974824, −4.44816896727005505075247525849, −3.33181489892417944509203766947, −1.22170529453993031958503762319,
1.45336961431069681295976527545, 2.73735879341228160890266913760, 3.85311957704317363610529591962, 5.19917521066221735508032589086, 6.07881318561331620899101621623, 6.84747155766038526284114738338, 7.85868480559283408897694564047, 9.115329336580309428186158593856, 10.09047398127685588403766588517, 10.83329556570078373280711029262