| 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 | \( 1 + (-13.4 - 11.2i)T + (16.8 + 95.5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.74581922313659891527713849328, −13.25724028140085861128823460496, −11.50286830841864173088940793279, −10.80067618363918704106065375433, −9.448622945062672558082236624108, −7.910325183757820215197374643656, −6.92131650133965409879110128299, −6.27331503240779594881168596759, −3.86957946349673115241725355650, −3.14228509481876831132742873618,
2.13048373333180802412150150202, 4.06590607101284889854178893659, 5.03566724736339800212697429513, 6.55574575181375588999787034321, 8.529409316511522522900186150535, 9.089130867482013196131638582693, 10.13751518198209976977803638343, 11.89355622538967513696847580277, 12.60063771862963386893011896803, 12.97785758725299493149196217483
