| L(s) = 1 | + (1.73 + 1.42i)2-s + (2.19 + 0.176i)3-s + (0.608 + 2.98i)4-s + (−0.992 + 0.120i)5-s + (3.56 + 3.42i)6-s + (−0.140 − 0.221i)7-s + (−1.08 + 2.07i)8-s + (1.81 + 0.294i)9-s + (−1.89 − 1.20i)10-s + (0.703 + 4.33i)11-s + (0.806 + 6.64i)12-s + (0.169 + 3.60i)13-s + (0.0710 − 0.585i)14-s + (−2.19 + 0.0885i)15-s + (0.758 − 0.323i)16-s + (4.64 − 2.93i)17-s + ⋯ |
| L(s) = 1 | + (1.23 + 1.00i)2-s + (1.26 + 0.102i)3-s + (0.304 + 1.49i)4-s + (−0.443 + 0.0539i)5-s + (1.45 + 1.39i)6-s + (−0.0530 − 0.0838i)7-s + (−0.384 + 0.733i)8-s + (0.603 + 0.0981i)9-s + (−0.600 − 0.379i)10-s + (0.212 + 1.30i)11-s + (0.232 + 1.91i)12-s + (0.0471 + 0.998i)13-s + (0.0189 − 0.156i)14-s + (−0.567 + 0.0228i)15-s + (0.189 − 0.0808i)16-s + (1.12 − 0.712i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 845 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.68124 + 3.13653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.68124 + 3.13653i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.992 - 0.120i)T \) |
| 13 | \( 1 + (-0.169 - 3.60i)T \) |
| good | 2 | \( 1 + (-1.73 - 1.42i)T + (0.400 + 1.95i)T^{2} \) |
| 3 | \( 1 + (-2.19 - 0.176i)T + (2.96 + 0.481i)T^{2} \) |
| 7 | \( 1 + (0.140 + 0.221i)T + (-3.00 + 6.32i)T^{2} \) |
| 11 | \( 1 + (-0.703 - 4.33i)T + (-10.4 + 3.48i)T^{2} \) |
| 17 | \( 1 + (-4.64 + 2.93i)T + (7.28 - 15.3i)T^{2} \) |
| 19 | \( 1 + (3.02 + 1.74i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.01 + 6.94i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.602 + 0.737i)T + (-5.80 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-0.258 + 1.04i)T + (-27.4 - 14.4i)T^{2} \) |
| 37 | \( 1 + (-1.51 + 0.439i)T + (31.2 - 19.7i)T^{2} \) |
| 41 | \( 1 + (0.847 - 10.4i)T + (-40.4 - 6.57i)T^{2} \) |
| 43 | \( 1 + (-2.41 + 8.35i)T + (-36.3 - 22.9i)T^{2} \) |
| 47 | \( 1 + (5.21 - 5.89i)T + (-5.66 - 46.6i)T^{2} \) |
| 53 | \( 1 + (4.50 + 2.36i)T + (30.1 + 43.6i)T^{2} \) |
| 59 | \( 1 + (-5.40 + 12.6i)T + (-40.8 - 42.5i)T^{2} \) |
| 61 | \( 1 + (0.308 - 7.64i)T + (-60.8 - 4.90i)T^{2} \) |
| 67 | \( 1 + (14.5 + 2.97i)T + (61.6 + 26.2i)T^{2} \) |
| 71 | \( 1 + (-1.81 + 0.860i)T + (44.9 - 54.9i)T^{2} \) |
| 73 | \( 1 + (0.512 + 0.194i)T + (54.6 + 48.4i)T^{2} \) |
| 79 | \( 1 + (-4.09 - 3.62i)T + (9.52 + 78.4i)T^{2} \) |
| 83 | \( 1 + (-10.0 + 6.94i)T + (29.4 - 77.6i)T^{2} \) |
| 89 | \( 1 + (-3.57 + 2.06i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.23 + 9.62i)T + (-26.9 - 93.1i)T^{2} \) |
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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
−10.15907810226817085233845818984, −9.427936991192219947149284380428, −8.409789473111728064340730576306, −7.70044291789831033497684244532, −6.98963090906880189990056708080, −6.21636920805657928943301799296, −4.76840027683969264919468744306, −4.26929502702700090648270314032, −3.34207299947215862097746232608, −2.22585993130507160331840552002,
1.43184555220920278425381998250, 2.74525440861307914861956065213, 3.48822196851322705922731277044, 3.91775884100632909736153429426, 5.43220826638348612564021267867, 6.03804305385433981410823838575, 7.79620479594799660280661946389, 8.146593455167266359349482865386, 9.143416685792859635202273019749, 10.24300595628231836606526160967
