| L(s) = 1 | + (0.766 − 0.642i)2-s + (0.939 + 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.613 − 3.47i)5-s + (0.939 − 0.342i)6-s + (−1.85 + 3.21i)7-s + (−0.500 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−2.70 − 2.27i)10-s + (2.64 + 4.58i)11-s + (0.499 − 0.866i)12-s + (0.213 − 0.0775i)13-s + (0.645 + 3.66i)14-s + (0.613 − 3.47i)15-s + (−0.939 − 0.342i)16-s + (−1.26 + 1.06i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (0.542 + 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.274 − 1.55i)5-s + (0.383 − 0.139i)6-s + (−0.702 + 1.21i)7-s + (−0.176 − 0.306i)8-s + (0.255 + 0.214i)9-s + (−0.855 − 0.717i)10-s + (0.797 + 1.38i)11-s + (0.144 − 0.250i)12-s + (0.0590 − 0.0215i)13-s + (0.172 + 0.978i)14-s + (0.158 − 0.898i)15-s + (−0.234 − 0.0855i)16-s + (−0.307 + 0.257i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33668 - 0.543879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33668 - 0.543879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.939 - 0.342i)T \) |
| 19 | \( 1 + (4.17 - 1.24i)T \) |
| good | 5 | \( 1 + (0.613 + 3.47i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (1.85 - 3.21i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.64 - 4.58i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.213 + 0.0775i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.26 - 1.06i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-1.50 + 8.54i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.0923 - 0.0775i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.56 - 2.70i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 + (6.67 + 2.43i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.929 + 5.27i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (1.92 + 1.61i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-1.03 + 5.84i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.167 + 0.140i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (0.273 - 1.55i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-11.8 - 9.95i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.235 + 1.33i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (2.27 + 0.829i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.69 - 0.979i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (0.960 - 1.66i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-11.4 + 4.15i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (-13.4 + 11.2i)T + (16.8 - 95.5i)T^{2} \) |
| 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
−12.97785758725299493149196217483, −12.60063771862963386893011896803, −11.89355622538967513696847580277, −10.13751518198209976977803638343, −9.089130867482013196131638582693, −8.529409316511522522900186150535, −6.55574575181375588999787034321, −5.03566724736339800212697429513, −4.06590607101284889854178893659, −2.13048373333180802412150150202,
3.14228509481876831132742873618, 3.86957946349673115241725355650, 6.27331503240779594881168596759, 6.92131650133965409879110128299, 7.910325183757820215197374643656, 9.448622945062672558082236624108, 10.80067618363918704106065375433, 11.50286830841864173088940793279, 13.25724028140085861128823460496, 13.74581922313659891527713849328
