L(s) = 1 | + (−0.354 + 0.935i)3-s + (−0.354 − 0.935i)4-s + (0.688 − 1.81i)7-s + (−0.748 − 0.663i)9-s + 12-s + 0.241·13-s + (−0.748 + 0.663i)16-s + (0.688 + 0.169i)19-s + (1.45 + 1.28i)21-s + (0.568 + 0.822i)25-s + (0.885 − 0.464i)27-s − 1.94·28-s + (0.136 − 1.12i)31-s + (−0.354 + 0.935i)36-s + (−1.71 + 0.902i)37-s + ⋯ |
L(s) = 1 | + (−0.354 + 0.935i)3-s + (−0.354 − 0.935i)4-s + (0.688 − 1.81i)7-s + (−0.748 − 0.663i)9-s + 12-s + 0.241·13-s + (−0.748 + 0.663i)16-s + (0.688 + 0.169i)19-s + (1.45 + 1.28i)21-s + (0.568 + 0.822i)25-s + (0.885 − 0.464i)27-s − 1.94·28-s + (0.136 − 1.12i)31-s + (−0.354 + 0.935i)36-s + (−1.71 + 0.902i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7466718784\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7466718784\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.354 - 0.935i)T \) |
| 157 | \( 1 + (0.354 - 0.935i)T \) |
good | 2 | \( 1 + (0.354 + 0.935i)T^{2} \) |
| 5 | \( 1 + (-0.568 - 0.822i)T^{2} \) |
| 7 | \( 1 + (-0.688 + 1.81i)T + (-0.748 - 0.663i)T^{2} \) |
| 11 | \( 1 + (0.354 - 0.935i)T^{2} \) |
| 13 | \( 1 - 0.241T + T^{2} \) |
| 17 | \( 1 + (-0.568 - 0.822i)T^{2} \) |
| 19 | \( 1 + (-0.688 - 0.169i)T + (0.885 + 0.464i)T^{2} \) |
| 23 | \( 1 + (-0.120 + 0.992i)T^{2} \) |
| 29 | \( 1 + (-0.568 + 0.822i)T^{2} \) |
| 31 | \( 1 + (-0.136 + 1.12i)T + (-0.970 - 0.239i)T^{2} \) |
| 37 | \( 1 + (1.71 - 0.902i)T + (0.568 - 0.822i)T^{2} \) |
| 41 | \( 1 + (-0.120 - 0.992i)T^{2} \) |
| 43 | \( 1 + (-0.530 - 1.39i)T + (-0.748 + 0.663i)T^{2} \) |
| 47 | \( 1 + (-0.568 - 0.822i)T^{2} \) |
| 53 | \( 1 + (-0.885 - 0.464i)T^{2} \) |
| 59 | \( 1 + (-0.885 + 0.464i)T^{2} \) |
| 61 | \( 1 + (1.10 + 0.271i)T + (0.885 + 0.464i)T^{2} \) |
| 67 | \( 1 + (-1.45 - 0.358i)T + (0.885 + 0.464i)T^{2} \) |
| 71 | \( 1 + (-0.885 - 0.464i)T^{2} \) |
| 73 | \( 1 + (0.627 - 1.65i)T + (-0.748 - 0.663i)T^{2} \) |
| 79 | \( 1 + (0.402 + 0.583i)T + (-0.354 + 0.935i)T^{2} \) |
| 83 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 89 | \( 1 + (0.970 + 0.239i)T^{2} \) |
| 97 | \( 1 + (-1.56 - 0.822i)T + (0.568 + 0.822i)T^{2} \) |
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
−11.01240711636923100664948386504, −10.23786177750731710031482070645, −9.724361485997275184318352875990, −8.638295474889987195446033291008, −7.45677460843016308980612125417, −6.37092088947012222282156281639, −5.21095813101250995910446385336, −4.51417395741628469178950658269, −3.58486516713060900435747031011, −1.16673175893700056709446876837,
2.04796592169459098043352833249, 3.10915271065040703393501263450, 4.88889500623154806050680027607, 5.63230892463637508235188073176, 6.80597061588173867907849122299, 7.79577330777980739183557071567, 8.643187898974223739536314360002, 9.020384734311007429835970846306, 10.72226748286046332327578004753, 11.73314584312763694221201408901