| L(s) = 1 | + (1.36 − 0.383i)2-s + (1.52 − 2.09i)3-s + (1.70 − 1.04i)4-s + (−2.19 + 0.434i)5-s + (1.26 − 3.43i)6-s + (−0.912 − 0.912i)7-s + (1.92 − 2.07i)8-s + (−1.14 − 3.51i)9-s + (−2.81 + 1.43i)10-s + (−1.23 + 0.631i)11-s + (0.411 − 5.16i)12-s + (0.633 + 1.94i)13-s + (−1.59 − 0.892i)14-s + (−2.42 + 5.25i)15-s + (1.82 − 3.56i)16-s + (−1.61 − 0.255i)17-s + ⋯ |
| L(s) = 1 | + (0.962 − 0.270i)2-s + (0.878 − 1.20i)3-s + (0.853 − 0.521i)4-s + (−0.980 + 0.194i)5-s + (0.517 − 1.40i)6-s + (−0.345 − 0.345i)7-s + (0.679 − 0.733i)8-s + (−0.381 − 1.17i)9-s + (−0.891 + 0.452i)10-s + (−0.373 + 0.190i)11-s + (0.118 − 1.48i)12-s + (0.175 + 0.540i)13-s + (−0.425 − 0.238i)14-s + (−0.626 + 1.35i)15-s + (0.455 − 0.890i)16-s + (−0.391 − 0.0619i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0664 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0664 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.81899 - 1.94417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.81899 - 1.94417i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 + 0.383i)T \) |
| 5 | \( 1 + (2.19 - 0.434i)T \) |
| good | 3 | \( 1 + (-1.52 + 2.09i)T + (-0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + (0.912 + 0.912i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.23 - 0.631i)T + (6.46 - 8.89i)T^{2} \) |
| 13 | \( 1 + (-0.633 - 1.94i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.61 + 0.255i)T + (16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-7.18 - 1.13i)T + (18.0 + 5.87i)T^{2} \) |
| 23 | \( 1 + (-2.93 - 5.75i)T + (-13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (1.61 - 0.255i)T + (27.5 - 8.96i)T^{2} \) |
| 31 | \( 1 + (-0.943 - 1.29i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.187 + 0.577i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.27 - 0.413i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + (4.36 - 0.690i)T + (44.6 - 14.5i)T^{2} \) |
| 53 | \( 1 + (4.72 - 6.49i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-7.46 - 3.80i)T + (34.6 + 47.7i)T^{2} \) |
| 61 | \( 1 + (1.67 - 0.852i)T + (35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (10.6 - 7.72i)T + (20.7 - 63.7i)T^{2} \) |
| 71 | \( 1 + (7.76 + 5.63i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-8.72 + 4.44i)T + (42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-8.72 - 6.34i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (8.63 + 11.8i)T + (-25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-4.68 + 14.4i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.74 - 11.0i)T + (-92.2 + 29.9i)T^{2} \) |
| 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
−11.57121446965205329402919863467, −10.32647687270820646384778523200, −9.115567414431341788029901892602, −7.79915616257369061705111630955, −7.29599683084387476305121658805, −6.55320639629756107038581261890, −5.05741565441000395967502113006, −3.65081640161813234987063569918, −2.93115216969670678299143187646, −1.43362480070965665603646311660,
2.89434198390912462990563072634, 3.44925101603750328523590748096, 4.55952201891626681614943476700, 5.30464601907200600939321495004, 6.79213063663353897420309403732, 7.961940325525892314370720805608, 8.578456840384970338326481611918, 9.675137804949605492508601569753, 10.74358445162805279404065558484, 11.51097995391136061738838040650
