| L(s) = 1 | + (−0.892 − 2.64i)2-s + (0.713 + 2.91i)3-s + (−3.03 + 2.30i)4-s + (−2.87 + 1.14i)5-s + (7.08 − 4.48i)6-s + (2.33 + 1.08i)7-s + (−0.437 − 0.296i)8-s + (−7.98 + 4.15i)9-s + (5.59 + 6.58i)10-s + (−5.90 + 1.64i)11-s + (−8.88 − 7.19i)12-s + (11.2 + 21.3i)13-s + (0.778 − 7.15i)14-s + (−5.38 − 7.55i)15-s + (−4.47 + 16.1i)16-s + (8.75 + 18.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.446 − 1.32i)2-s + (0.237 + 0.971i)3-s + (−0.758 + 0.576i)4-s + (−0.574 + 0.228i)5-s + (1.18 − 0.748i)6-s + (0.333 + 0.154i)7-s + (−0.0547 − 0.0371i)8-s + (−0.886 + 0.461i)9-s + (0.559 + 0.658i)10-s + (−0.537 + 0.149i)11-s + (−0.740 − 0.599i)12-s + (0.869 + 1.63i)13-s + (0.0555 − 0.511i)14-s + (−0.358 − 0.503i)15-s + (−0.279 + 1.00i)16-s + (0.514 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.771389 + 0.365732i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.771389 + 0.365732i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.713 - 2.91i)T \) |
| 59 | \( 1 + (5.88 - 58.7i)T \) |
| good | 2 | \( 1 + (0.892 + 2.64i)T + (-3.18 + 2.42i)T^{2} \) |
| 5 | \( 1 + (2.87 - 1.14i)T + (18.1 - 17.1i)T^{2} \) |
| 7 | \( 1 + (-2.33 - 1.08i)T + (31.7 + 37.3i)T^{2} \) |
| 11 | \( 1 + (5.90 - 1.64i)T + (103. - 62.3i)T^{2} \) |
| 13 | \( 1 + (-11.2 - 21.3i)T + (-94.8 + 139. i)T^{2} \) |
| 17 | \( 1 + (-8.75 - 18.9i)T + (-187. + 220. i)T^{2} \) |
| 19 | \( 1 + (-1.82 - 0.402i)T + (327. + 151. i)T^{2} \) |
| 23 | \( 1 + (21.5 - 3.52i)T + (501. - 168. i)T^{2} \) |
| 29 | \( 1 + (-2.19 + 6.51i)T + (-669. - 508. i)T^{2} \) |
| 31 | \( 1 + (-44.5 + 9.81i)T + (872. - 403. i)T^{2} \) |
| 37 | \( 1 + (32.5 + 47.9i)T + (-506. + 1.27e3i)T^{2} \) |
| 41 | \( 1 + (-4.66 - 0.765i)T + (1.59e3 + 536. i)T^{2} \) |
| 43 | \( 1 + (12.9 - 46.5i)T + (-1.58e3 - 953. i)T^{2} \) |
| 47 | \( 1 + (-22.9 - 9.15i)T + (1.60e3 + 1.51e3i)T^{2} \) |
| 53 | \( 1 + (4.66 + 3.96i)T + (454. + 2.77e3i)T^{2} \) |
| 61 | \( 1 + (-3.40 + 1.14i)T + (2.96e3 - 2.25e3i)T^{2} \) |
| 67 | \( 1 + (-27.0 + 39.9i)T + (-1.66e3 - 4.17e3i)T^{2} \) |
| 71 | \( 1 + (32.5 + 12.9i)T + (3.65e3 + 3.46e3i)T^{2} \) |
| 73 | \( 1 + (-20.7 - 2.25i)T + (5.20e3 + 1.14e3i)T^{2} \) |
| 79 | \( 1 + (82.7 + 49.7i)T + (2.92e3 + 5.51e3i)T^{2} \) |
| 83 | \( 1 + (-104. - 5.66i)T + (6.84e3 + 744. i)T^{2} \) |
| 89 | \( 1 + (-6.55 + 19.4i)T + (-6.30e3 - 4.79e3i)T^{2} \) |
| 97 | \( 1 + (83.6 - 9.09i)T + (9.18e3 - 2.02e3i)T^{2} \) |
| 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
−12.04135585375474235134557101688, −11.45015740970696410626435393689, −10.66041878743297028194764841258, −9.834036455323362060852107648684, −8.874039492553844448591792658748, −7.988472026352566660728791573933, −6.09201000757209297133399995610, −4.31445594802235958264084365306, −3.51438181474313508731999345223, −1.97955214246138884484815084204,
0.57533168057218859014601428482, 3.05062108283305763339708461325, 5.20941899552980508480497604061, 6.19801151300222428408638986733, 7.36718395292165937943979958335, 8.124823989606157384255101705674, 8.495434393645654064384158654550, 10.10030399374119927888884593796, 11.55600097776365091250536165670, 12.37465085817041439061992680726
