| L(s) = 1 | + (0.535 − 0.844i)2-s + (−0.224 + 0.210i)3-s + (−0.425 − 0.904i)4-s + (1.43 + 1.71i)5-s + (0.0576 + 0.302i)6-s + (2.26 − 1.64i)7-s + (−0.992 − 0.125i)8-s + (−0.182 + 2.89i)9-s + (2.21 − 0.289i)10-s + (1.56 − 2.47i)11-s + (0.286 + 0.113i)12-s + (−0.0503 + 0.800i)13-s + (−0.175 − 2.79i)14-s + (−0.683 − 0.0833i)15-s + (−0.637 + 0.770i)16-s + (1.59 − 3.38i)17-s + ⋯ |
| L(s) = 1 | + (0.378 − 0.597i)2-s + (−0.129 + 0.121i)3-s + (−0.212 − 0.452i)4-s + (0.640 + 0.767i)5-s + (0.0235 + 0.123i)6-s + (0.856 − 0.622i)7-s + (−0.350 − 0.0443i)8-s + (−0.0608 + 0.966i)9-s + (0.701 − 0.0916i)10-s + (0.472 − 0.745i)11-s + (0.0826 + 0.0327i)12-s + (−0.0139 + 0.221i)13-s + (−0.0470 − 0.747i)14-s + (−0.176 − 0.0215i)15-s + (−0.159 + 0.192i)16-s + (0.386 − 0.820i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59279 - 0.440780i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59279 - 0.440780i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.535 + 0.844i)T \) |
| 5 | \( 1 + (-1.43 - 1.71i)T \) |
| good | 3 | \( 1 + (0.224 - 0.210i)T + (0.188 - 2.99i)T^{2} \) |
| 7 | \( 1 + (-2.26 + 1.64i)T + (2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (-1.56 + 2.47i)T + (-4.68 - 9.95i)T^{2} \) |
| 13 | \( 1 + (0.0503 - 0.800i)T + (-12.8 - 1.62i)T^{2} \) |
| 17 | \( 1 + (-1.59 + 3.38i)T + (-10.8 - 13.0i)T^{2} \) |
| 19 | \( 1 + (1.03 + 0.973i)T + (1.19 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-1.09 - 0.279i)T + (20.1 + 11.0i)T^{2} \) |
| 29 | \( 1 + (5.84 + 3.21i)T + (15.5 + 24.4i)T^{2} \) |
| 31 | \( 1 + (1.41 - 3.00i)T + (-19.7 - 23.8i)T^{2} \) |
| 37 | \( 1 + (0.611 - 0.739i)T + (-6.93 - 36.3i)T^{2} \) |
| 41 | \( 1 + (9.81 - 2.51i)T + (35.9 - 19.7i)T^{2} \) |
| 43 | \( 1 + (2.17 - 6.69i)T + (-34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-0.00133 + 0.000169i)T + (45.5 - 11.6i)T^{2} \) |
| 53 | \( 1 + (0.0865 - 0.453i)T + (-49.2 - 19.5i)T^{2} \) |
| 59 | \( 1 + (-2.38 - 0.942i)T + (43.0 + 40.3i)T^{2} \) |
| 61 | \( 1 + (8.85 + 2.27i)T + (53.4 + 29.3i)T^{2} \) |
| 67 | \( 1 + (-13.7 + 7.56i)T + (35.9 - 56.5i)T^{2} \) |
| 71 | \( 1 + (7.19 - 0.908i)T + (68.7 - 17.6i)T^{2} \) |
| 73 | \( 1 + (-6.61 + 2.61i)T + (53.2 - 49.9i)T^{2} \) |
| 79 | \( 1 + (-0.880 + 0.827i)T + (4.96 - 78.8i)T^{2} \) |
| 83 | \( 1 + (8.57 + 8.05i)T + (5.21 + 82.8i)T^{2} \) |
| 89 | \( 1 + (12.7 - 5.06i)T + (64.8 - 60.9i)T^{2} \) |
| 97 | \( 1 + (-3.88 - 2.13i)T + (51.9 + 81.8i)T^{2} \) |
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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
−11.56495057776503697472324818000, −11.13502037994903082040846471658, −10.35514925303828664888860814744, −9.379030993367521617813435996227, −8.035650965159655880544700257780, −6.89550702948546037789830644663, −5.62923393138830022327867952016, −4.64430493143997512428602689716, −3.20249935050435266454586863130, −1.75639606881556609430205498316,
1.77422490944436388862547003588, 3.86477203132389225146019522619, 5.14108458743105835846528557749, 5.88873503961013960992441174378, 7.01505114755636065885677581283, 8.334484218082638723524367491912, 9.016296929184226101429609669675, 10.02362448855146131460407981805, 11.53061210119202131789310469348, 12.37986893109297298879568308690
