L(s) = 1 | − 0.201i·2-s + (−2.12 + 2.11i)3-s + 3.95·4-s − 6.21i·5-s + (0.425 + 0.428i)6-s − 0.661·7-s − 1.60i·8-s + (0.0531 − 8.99i)9-s − 1.25·10-s − 5.79i·11-s + (−8.42 + 8.37i)12-s + 3.06·13-s + 0.133i·14-s + (13.1 + 13.2i)15-s + 15.5·16-s − 12.8i·17-s + ⋯ |
L(s) = 1 | − 0.100i·2-s + (−0.709 + 0.705i)3-s + 0.989·4-s − 1.24i·5-s + (0.0709 + 0.0714i)6-s − 0.0945·7-s − 0.200i·8-s + (0.00590 − 0.999i)9-s − 0.125·10-s − 0.526i·11-s + (−0.702 + 0.697i)12-s + 0.235·13-s + 0.00951i·14-s + (0.876 + 0.881i)15-s + 0.969·16-s − 0.754i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.709i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.705 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.33232 - 0.554176i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33232 - 0.554176i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.12 - 2.11i)T \) |
| 59 | \( 1 - 7.68iT \) |
good | 2 | \( 1 + 0.201iT - 4T^{2} \) |
| 5 | \( 1 + 6.21iT - 25T^{2} \) |
| 7 | \( 1 + 0.661T + 49T^{2} \) |
| 11 | \( 1 + 5.79iT - 121T^{2} \) |
| 13 | \( 1 - 3.06T + 169T^{2} \) |
| 17 | \( 1 + 12.8iT - 289T^{2} \) |
| 19 | \( 1 - 24.1T + 361T^{2} \) |
| 23 | \( 1 + 10.5iT - 529T^{2} \) |
| 29 | \( 1 - 3.55iT - 841T^{2} \) |
| 31 | \( 1 - 23.2T + 961T^{2} \) |
| 37 | \( 1 + 63.0T + 1.36e3T^{2} \) |
| 41 | \( 1 - 25.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 26.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 4.63iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 39.5iT - 2.80e3T^{2} \) |
| 61 | \( 1 - 48.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 35.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 103. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 76.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 26.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + 52.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.05iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 106.T + 9.40e3T^{2} \) |
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
−12.00162295417933949014040747372, −11.54153463881595870153338680640, −10.42268671565089812268866145147, −9.474091445370993579568444557585, −8.391542051738397213019661207078, −6.93195238621411098886046424789, −5.74339873526850795257067214963, −4.83888187783805850488594155057, −3.28309049174242978690687048305, −1.01783959824420087038746807049,
1.82133539160485636826796782056, 3.23046664924610284579866114528, 5.43002109944633719173205908199, 6.53964483585477533711770753257, 7.07926376164753467412226107989, 8.026378761693394569186010551621, 10.03810495676138561051533851683, 10.76828434222080281877915396580, 11.59227451627383686453106119589, 12.29328235239802162329307422403