L(s) = 1 | − 1.73·2-s + 1.73·3-s + 0.999·4-s − 2.99·6-s + (1.73 + 2i)7-s + 1.73·8-s + 2.99·9-s − 3.46i·11-s + 1.73·12-s + (−2.99 − 3.46i)14-s − 5·16-s + 6i·17-s − 5.19·18-s − 3.46i·19-s + (2.99 + 3.46i)21-s + 5.99i·22-s + ⋯ |
L(s) = 1 | − 1.22·2-s + 1.00·3-s + 0.499·4-s − 1.22·6-s + (0.654 + 0.755i)7-s + 0.612·8-s + 0.999·9-s − 1.04i·11-s + 0.500·12-s + (−0.801 − 0.925i)14-s − 1.25·16-s + 1.45i·17-s − 1.22·18-s − 0.794i·19-s + (0.654 + 0.755i)21-s + 1.27i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 - 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15773 + 0.145520i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15773 + 0.145520i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.73 - 2i)T \) |
good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 3.46T + 23T^{2} \) |
| 29 | \( 1 - 6.92iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−10.74468419557150499902276473547, −9.734371329967574109255276923591, −8.800401873638799884185958784555, −8.553375506640330609193478756077, −7.76705470977258941458935466993, −6.70267301208308914027839350482, −5.27944241422446973205677357206, −3.97900958360396786414068742996, −2.57606205125438410781092667339, −1.34277948835798500127276589344,
1.18870888224860901074469397589, 2.42076348425047111622674158741, 4.07421525887621953210749049990, 4.89855763857709975144732793745, 6.95027573315165821563669462409, 7.50528039832052153106196428743, 8.148946026181638605974642358321, 9.128940917908522275627797698986, 9.770338957526745238007914661872, 10.40335084660920893921396217859