L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.707 + 0.707i)3-s + 1.00i·4-s + (−0.253 + 2.22i)5-s + 1.00·6-s + (−2.47 + 2.47i)7-s + (0.707 − 0.707i)8-s − 1.00i·9-s + (1.75 − 1.39i)10-s + 2.74·11-s + (−0.707 − 0.707i)12-s + (1.20 − 1.20i)13-s + 3.50·14-s + (−1.39 − 1.75i)15-s − 1.00·16-s + (−4.87 + 4.87i)17-s + ⋯ |
L(s) = 1 | + (−0.499 − 0.499i)2-s + (−0.408 + 0.408i)3-s + 0.500i·4-s + (−0.113 + 0.993i)5-s + 0.408·6-s + (−0.935 + 0.935i)7-s + (0.250 − 0.250i)8-s − 0.333i·9-s + (0.553 − 0.440i)10-s + 0.827·11-s + (−0.204 − 0.204i)12-s + (0.333 − 0.333i)13-s + 0.935·14-s + (−0.359 − 0.451i)15-s − 0.250·16-s + (−1.18 + 1.18i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0347170 + 0.335798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0347170 + 0.335798i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.253 - 2.22i)T \) |
| 19 | \( 1 + (1.94 + 3.90i)T \) |
good | 7 | \( 1 + (2.47 - 2.47i)T - 7iT^{2} \) |
| 11 | \( 1 - 2.74T + 11T^{2} \) |
| 13 | \( 1 + (-1.20 + 1.20i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.87 - 4.87i)T - 17iT^{2} \) |
| 23 | \( 1 + (-0.0321 - 0.0321i)T + 23iT^{2} \) |
| 29 | \( 1 + 6.50T + 29T^{2} \) |
| 31 | \( 1 + 6.50iT - 31T^{2} \) |
| 37 | \( 1 + (4.58 + 4.58i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.96iT - 41T^{2} \) |
| 43 | \( 1 + (-5.39 - 5.39i)T + 43iT^{2} \) |
| 47 | \( 1 + (-3.66 + 3.66i)T - 47iT^{2} \) |
| 53 | \( 1 + (8.97 - 8.97i)T - 53iT^{2} \) |
| 59 | \( 1 + 4.42T + 59T^{2} \) |
| 61 | \( 1 + 2.95T + 61T^{2} \) |
| 67 | \( 1 + (7.00 + 7.00i)T + 67iT^{2} \) |
| 71 | \( 1 - 5.56iT - 71T^{2} \) |
| 73 | \( 1 + (2.19 + 2.19i)T + 73iT^{2} \) |
| 79 | \( 1 - 0.225T + 79T^{2} \) |
| 83 | \( 1 + (3.87 + 3.87i)T + 83iT^{2} \) |
| 89 | \( 1 - 9.13T + 89T^{2} \) |
| 97 | \( 1 + (-8.76 - 8.76i)T + 97iT^{2} \) |
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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
−11.03764490537274373075150873412, −10.43388350257403678961707544781, −9.306122883350531058024011912942, −8.982238663800216063094836818911, −7.61764982393746041993674641686, −6.40149704841424157770650464755, −6.05272938760906316874465102481, −4.25863970468387340597851830335, −3.32465009904882975528501660196, −2.20155277323086302692439126398,
0.23468687304048084947566864387, 1.60803696079533543218176037815, 3.76378153345659601048576948569, 4.77860042346309242565187169350, 5.98240595059851871494970012418, 6.81116567051728899799482403202, 7.45893944108351912983940097941, 8.711946488219374594093827103545, 9.244916884726688845082416293918, 10.20194332283918183695827840716