| L(s) = 1 | + (−0.795 − 2.18i)2-s + (−1.13 − 0.199i)3-s + (−2.60 + 2.18i)4-s + (−2.16 − 0.545i)5-s + (0.463 + 2.62i)6-s + (−0.124 + 0.0716i)7-s + (2.82 + 1.63i)8-s + (−1.57 − 0.574i)9-s + (0.531 + 5.17i)10-s + (1.40 − 2.43i)11-s + (3.38 − 1.95i)12-s + (1.68 − 0.296i)13-s + (0.255 + 0.214i)14-s + (2.34 + 1.04i)15-s + (0.135 − 0.766i)16-s + (−1.21 − 3.34i)17-s + ⋯ |
| L(s) = 1 | + (−0.562 − 1.54i)2-s + (−0.653 − 0.115i)3-s + (−1.30 + 1.09i)4-s + (−0.969 − 0.244i)5-s + (0.189 + 1.07i)6-s + (−0.0469 + 0.0270i)7-s + (0.999 + 0.576i)8-s + (−0.526 − 0.191i)9-s + (0.168 + 1.63i)10-s + (0.424 − 0.735i)11-s + (0.977 − 0.564i)12-s + (0.466 − 0.0822i)13-s + (0.0682 + 0.0572i)14-s + (0.605 + 0.271i)15-s + (0.0338 − 0.191i)16-s + (−0.295 − 0.811i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.851 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0934245 + 0.330473i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0934245 + 0.330473i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.16 + 0.545i)T \) |
| 19 | \( 1 + (2.82 + 3.32i)T \) |
| good | 2 | \( 1 + (0.795 + 2.18i)T + (-1.53 + 1.28i)T^{2} \) |
| 3 | \( 1 + (1.13 + 0.199i)T + (2.81 + 1.02i)T^{2} \) |
| 7 | \( 1 + (0.124 - 0.0716i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.40 + 2.43i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.68 + 0.296i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (1.21 + 3.34i)T + (-13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (4.62 + 5.51i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-7.11 - 2.58i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (2.42 + 4.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 9.49iT - 37T^{2} \) |
| 41 | \( 1 + (0.0325 - 0.184i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.06 + 8.42i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (1.25 - 3.44i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (1.05 + 1.25i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (5.04 - 1.83i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-7.74 + 6.50i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.05 + 5.64i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.414 - 0.347i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (10.3 + 1.81i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-2.01 + 11.4i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (6.57 - 3.79i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.251 + 1.42i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (3.01 + 8.27i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−12.78313454413630028668359281991, −11.92533915492405531045825426817, −11.33218867941469622324226333525, −10.57674820508755290533984998853, −9.029016897825398763586696131057, −8.327627428673336485475787526172, −6.41982167743679074234118235590, −4.46690924774267269335816666339, −3.02670088330995583387424065682, −0.53370836170152313020998354182,
4.21917027016479678133232366583, 5.74140552947693176693866203063, 6.69720628213635741786607086567, 7.86365624702132857220905539972, 8.690683806742407482714521644416, 10.15576523021330999787907182604, 11.35479581442352805110791594354, 12.42341120152037095891829879257, 14.14066946471042780514928711565, 14.86506764642562215279133511133
