| L(s) = 1 | + (1.41 − 0.0658i)2-s + (0.141 − 1.72i)3-s + (1.99 − 0.186i)4-s + (−0.635 + 0.521i)5-s + (0.0863 − 2.44i)6-s + (−2.61 − 3.90i)7-s + (2.80 − 0.394i)8-s + (−2.95 − 0.489i)9-s + (−0.862 + 0.778i)10-s + (1.17 − 0.357i)11-s + (−0.0392 − 3.46i)12-s + (0.549 + 0.450i)13-s + (−3.94 − 5.35i)14-s + (0.809 + 1.17i)15-s + (3.93 − 0.741i)16-s + (0.884 + 2.13i)17-s + ⋯ |
| L(s) = 1 | + (0.998 − 0.0465i)2-s + (0.0817 − 0.996i)3-s + (0.995 − 0.0930i)4-s + (−0.284 + 0.233i)5-s + (0.0352 − 0.999i)6-s + (−0.987 − 1.47i)7-s + (0.990 − 0.139i)8-s + (−0.986 − 0.163i)9-s + (−0.272 + 0.246i)10-s + (0.355 − 0.107i)11-s + (−0.0113 − 0.999i)12-s + (0.152 + 0.125i)13-s + (−1.05 − 1.42i)14-s + (0.209 + 0.302i)15-s + (0.982 − 0.185i)16-s + (0.214 + 0.518i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0969 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0969 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.67380 - 1.51861i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.67380 - 1.51861i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 0.0658i)T \) |
| 3 | \( 1 + (-0.141 + 1.72i)T \) |
| good | 5 | \( 1 + (0.635 - 0.521i)T + (0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (2.61 + 3.90i)T + (-2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (-1.17 + 0.357i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-0.549 - 0.450i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-0.884 - 2.13i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-3.14 + 0.309i)T + (18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-5.92 + 1.17i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-3.75 - 1.14i)T + (24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (5.90 - 5.90i)T - 31iT^{2} \) |
| 37 | \( 1 + (4.73 + 0.466i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-8.25 + 1.64i)T + (37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (-0.894 + 1.67i)T + (-23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (2.26 + 5.47i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (6.24 - 1.89i)T + (44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (2.15 - 1.76i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (6.77 + 12.6i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (1.16 - 0.623i)T + (37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (-7.24 - 10.8i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (-5.44 - 3.63i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-7.26 - 3.00i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-3.62 + 0.356i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (3.36 - 16.9i)T + (-82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (-3.90 - 3.90i)T + 97iT^{2} \) |
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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
−11.15764565217286662518196718802, −10.68642705946829772091185503894, −9.349288119325118345408305095253, −7.85962561473351790513308885443, −6.96213162768241732743914822555, −6.66701427039295253604974990985, −5.34303024147878515548160914680, −3.78787805044493949474471148109, −3.11055439080229569132844876571, −1.23500806113289197046111554227,
2.64851162452358148783853377969, 3.44538858089717435575942904668, 4.69726746058970728458956542962, 5.60059493514646986938122434636, 6.36895136801200631753224116514, 7.80550787175772137317932591460, 9.049610231689615340384557015280, 9.628530272128359423862561599875, 10.84844507342407815486402310395, 11.75458928403301556331006667110
