L(s) = 1 | + (0.618 + 1.90i)2-s + (0.809 + 0.587i)3-s + (−1.61 + 1.17i)4-s + (1.23 − 3.80i)5-s + (−0.618 + 1.90i)6-s + (0.809 − 0.587i)7-s + (0.309 + 0.951i)9-s + 7.99·10-s − 1.99·12-s + (0.618 + 1.90i)13-s + (1.61 + 1.17i)14-s + (3.23 − 2.35i)15-s + (−1.23 + 3.80i)16-s + (−1.23 + 3.80i)17-s + (−1.61 + 1.17i)18-s + (−2.42 − 1.76i)19-s + ⋯ |
L(s) = 1 | + (0.437 + 1.34i)2-s + (0.467 + 0.339i)3-s + (−0.809 + 0.587i)4-s + (0.552 − 1.70i)5-s + (−0.252 + 0.776i)6-s + (0.305 − 0.222i)7-s + (0.103 + 0.317i)9-s + 2.52·10-s − 0.577·12-s + (0.171 + 0.527i)13-s + (0.432 + 0.314i)14-s + (0.835 − 0.607i)15-s + (−0.309 + 0.951i)16-s + (−0.299 + 0.922i)17-s + (−0.381 + 0.277i)18-s + (−0.556 − 0.404i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.71846 + 1.31206i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.71846 + 1.31206i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.618 - 1.90i)T + (-1.61 + 1.17i)T^{2} \) |
| 5 | \( 1 + (-1.23 + 3.80i)T + (-4.04 - 2.93i)T^{2} \) |
| 7 | \( 1 + (-0.809 + 0.587i)T + (2.16 - 6.65i)T^{2} \) |
| 13 | \( 1 + (-0.618 - 1.90i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.23 - 3.80i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (2.42 + 1.76i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 + (-4.85 + 3.52i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.54 + 4.75i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (2.42 - 1.76i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (1.61 + 1.17i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + (1.61 + 1.17i)T + (14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.85 - 5.70i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.09 + 5.87i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.927 - 2.85i)T + (-49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + T + 67T^{2} \) |
| 71 | \( 1 + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (8.89 - 6.46i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (3.39 + 10.4i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (1.85 - 5.70i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 12T + 89T^{2} \) |
| 97 | \( 1 + (-1.54 - 4.75i)T + (-78.4 + 57.0i)T^{2} \) |
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
−11.82786304164387173035538290214, −10.51114898130113743638293183219, −9.361857442229668437196602957612, −8.535296350530508220496109766154, −8.095960379650779406713820007232, −6.72051016256054584965644855639, −5.72102850617594003816902025012, −4.76120497561502996147941142752, −4.19009347778286151039189145019, −1.77342867087038813933050586413,
1.83813456029861339633822950396, 2.82630080265690501179832189184, 3.48844896291684355147802204782, 5.10108224300748058215309484366, 6.56529712998703429070053798983, 7.28162140044125638184862420894, 8.642366406757221136717410721457, 9.923985517587944492859195644037, 10.43355978283587790423559035483, 11.24868248685575492083375035486