L(s) = 1 | + (−0.0747 − 0.997i)2-s + (1.52 + 0.471i)3-s + (−0.988 + 0.149i)4-s + (−0.144 + 0.133i)5-s + (0.355 − 1.55i)6-s + (2.44 − 1.00i)7-s + (0.222 + 0.974i)8-s + (−0.364 − 0.248i)9-s + (0.144 + 0.133i)10-s + (−3.03 + 2.07i)11-s + (−1.58 − 0.238i)12-s + (−1.91 − 0.921i)13-s + (−1.18 − 2.36i)14-s + (−0.283 + 0.136i)15-s + (0.955 − 0.294i)16-s + (1.42 + 3.63i)17-s + ⋯ |
L(s) = 1 | + (−0.0528 − 0.705i)2-s + (0.882 + 0.272i)3-s + (−0.494 + 0.0745i)4-s + (−0.0645 + 0.0598i)5-s + (0.145 − 0.636i)6-s + (0.925 − 0.379i)7-s + (0.0786 + 0.344i)8-s + (−0.121 − 0.0827i)9-s + (0.0456 + 0.0423i)10-s + (−0.915 + 0.624i)11-s + (−0.456 − 0.0688i)12-s + (−0.530 − 0.255i)13-s + (−0.316 − 0.632i)14-s + (−0.0732 + 0.0352i)15-s + (0.238 − 0.0736i)16-s + (0.346 + 0.882i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.794 + 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.11289 - 0.376981i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.11289 - 0.376981i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0747 + 0.997i)T \) |
| 7 | \( 1 + (-2.44 + 1.00i)T \) |
good | 3 | \( 1 + (-1.52 - 0.471i)T + (2.47 + 1.68i)T^{2} \) |
| 5 | \( 1 + (0.144 - 0.133i)T + (0.373 - 4.98i)T^{2} \) |
| 11 | \( 1 + (3.03 - 2.07i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (1.91 + 0.921i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + (-1.42 - 3.63i)T + (-12.4 + 11.5i)T^{2} \) |
| 19 | \( 1 + (2.48 + 4.29i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.79 - 4.56i)T + (-16.8 - 15.6i)T^{2} \) |
| 29 | \( 1 + (0.499 - 0.625i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (0.00876 - 0.0151i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.77 - 1.32i)T + (35.3 + 10.9i)T^{2} \) |
| 41 | \( 1 + (0.431 + 1.88i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-2.17 + 9.53i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (0.497 + 6.63i)T + (-46.4 + 7.00i)T^{2} \) |
| 53 | \( 1 + (-1.69 + 0.256i)T + (50.6 - 15.6i)T^{2} \) |
| 59 | \( 1 + (10.5 + 9.74i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (-9.87 - 1.48i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (3.63 - 6.29i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.28 - 7.88i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (0.0400 - 0.534i)T + (-72.1 - 10.8i)T^{2} \) |
| 79 | \( 1 + (-3.58 - 6.21i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.11 + 2.46i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (-11.0 - 7.55i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + 14.8T + 97T^{2} \) |
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
−13.76629786757351256948879513566, −12.85640283888682946533121170058, −11.58568371256239199589915343508, −10.57103866025929649205023434608, −9.564679835587160551822393867829, −8.391420866762324497520707331449, −7.51007340057050973522263166545, −5.22700500501401340544805151004, −3.83713018786563063392669706203, −2.29026062402909445993259642603,
2.57528810417995109468605858266, 4.65819487938708336282582912873, 5.99140668536068573652854551172, 7.82750710136583091619167677143, 8.114101517020926541386122509758, 9.319400407338978333495331973993, 10.75326708470008984576735586985, 12.11240434158948459943096651397, 13.32344971995344446645363933403, 14.36619206407316825891850330712