| L(s) = 1 | + (0.342 + 1.37i)2-s + (−0.143 − 1.27i)3-s + (−1.76 + 0.940i)4-s + (−2.93 − 1.41i)5-s + (1.69 − 0.632i)6-s + (−2.05 − 1.63i)7-s + (−1.89 − 2.09i)8-s + (1.32 − 0.302i)9-s + (0.933 − 4.51i)10-s + (−0.595 − 0.947i)11-s + (1.44 + 2.11i)12-s + (0.439 − 1.92i)13-s + (1.54 − 3.38i)14-s + (−1.37 + 3.93i)15-s + (2.22 − 3.32i)16-s + (−1.10 + 1.10i)17-s + ⋯ |
| L(s) = 1 | + (0.242 + 0.970i)2-s + (−0.0827 − 0.734i)3-s + (−0.882 + 0.470i)4-s + (−1.31 − 0.632i)5-s + (0.692 − 0.258i)6-s + (−0.776 − 0.619i)7-s + (−0.670 − 0.742i)8-s + (0.442 − 0.100i)9-s + (0.295 − 1.42i)10-s + (−0.179 − 0.285i)11-s + (0.418 + 0.609i)12-s + (0.121 − 0.533i)13-s + (0.412 − 0.903i)14-s + (−0.355 + 1.01i)15-s + (0.557 − 0.830i)16-s + (−0.267 + 0.267i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0497 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0497 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.393891 - 0.413986i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.393891 - 0.413986i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.342 - 1.37i)T \) |
| 29 | \( 1 + (5.02 - 1.94i)T \) |
| good | 3 | \( 1 + (0.143 + 1.27i)T + (-2.92 + 0.667i)T^{2} \) |
| 5 | \( 1 + (2.93 + 1.41i)T + (3.11 + 3.90i)T^{2} \) |
| 7 | \( 1 + (2.05 + 1.63i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (0.595 + 0.947i)T + (-4.77 + 9.91i)T^{2} \) |
| 13 | \( 1 + (-0.439 + 1.92i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (1.10 - 1.10i)T - 17iT^{2} \) |
| 19 | \( 1 + (5.21 + 0.588i)T + (18.5 + 4.22i)T^{2} \) |
| 23 | \( 1 + (-1.49 - 3.09i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (1.33 + 3.82i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (-9.43 - 5.93i)T + (16.0 + 33.3i)T^{2} \) |
| 41 | \( 1 + (-5.66 - 5.66i)T + 41iT^{2} \) |
| 43 | \( 1 + (-3.71 + 10.6i)T + (-33.6 - 26.8i)T^{2} \) |
| 47 | \( 1 + (2.88 + 4.58i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (-2.87 + 5.97i)T + (-33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + (-4.75 + 0.535i)T + (59.4 - 13.5i)T^{2} \) |
| 67 | \( 1 + (-2.51 + 0.573i)T + (60.3 - 29.0i)T^{2} \) |
| 71 | \( 1 + (-0.174 + 0.766i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (4.03 + 1.41i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (-4.55 + 7.24i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (-1.41 - 1.77i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (12.6 - 4.41i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (-1.45 + 12.9i)T + (-94.5 - 21.5i)T^{2} \) |
| 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
−12.40295046820177947068021728184, −11.20828602866942956349930022070, −9.776625959771720094834150470718, −8.558568342039511006594737521200, −7.74865769136030997396259723527, −7.04816116672243660999022580297, −6.01048021905907740920991724627, −4.46731112612493677100500423114, −3.61436601490447737884649293239, −0.43851518060630810878975347124,
2.62486168250982086184732505704, 3.88929273609737931664580036949, 4.53901845965118196823006843424, 6.16773334593415667662919460144, 7.55060876644163241806978286216, 8.930877752850615654993719750253, 9.687448987954821657440230082719, 10.84918972854123045994062411812, 11.19722695901838786627062162071, 12.46425913474089757590533814849
