L(s) = 1 | + (−0.5 + 0.363i)3-s + (−0.809 + 0.587i)4-s + 5-s + (−0.190 + 0.587i)9-s + (0.309 + 0.951i)11-s + (0.190 − 0.587i)12-s + (−0.5 + 0.363i)15-s + (0.309 − 0.951i)16-s + (−0.809 + 0.587i)20-s + (−0.5 − 1.53i)23-s + 25-s + (−0.309 − 0.951i)27-s + (−0.5 − 0.363i)31-s + (−0.5 − 0.363i)33-s + (−0.190 − 0.587i)36-s + (0.618 − 1.90i)37-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.363i)3-s + (−0.809 + 0.587i)4-s + 5-s + (−0.190 + 0.587i)9-s + (0.309 + 0.951i)11-s + (0.190 − 0.587i)12-s + (−0.5 + 0.363i)15-s + (0.309 − 0.951i)16-s + (−0.809 + 0.587i)20-s + (−0.5 − 1.53i)23-s + 25-s + (−0.309 − 0.951i)27-s + (−0.5 − 0.363i)31-s + (−0.5 − 0.363i)33-s + (−0.190 − 0.587i)36-s + (0.618 − 1.90i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6407678222\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6407678222\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 11 | \( 1 + (-0.309 - 0.951i)T \) |
good | 2 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 3 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.190 + 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 97 | \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \) |
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
−12.51674691152367709486189549484, −11.25894575156661882219616675209, −10.23090403948641184755904119919, −9.545247778330112464805455892436, −8.593747755220393752009448148288, −7.44732856757525946572974314846, −6.11508965505815348293830258521, −5.04570630052482461128376052956, −4.20808554678611797113438563828, −2.37703672066448835708596670930,
1.36827022788482007749117835448, 3.49908692231684940212974283131, 5.15511674058048184186949946256, 5.89767346726350481121266210590, 6.66195618170165305168313605544, 8.333079195952550185148115903391, 9.284902059428383652897460021017, 9.925560613076796172392326680578, 11.03426385752748825268808252932, 11.93902189655513318252839885734