| L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.199 − 0.613i)3-s + (0.309 + 0.951i)4-s + (1.52 + 1.63i)5-s + (−0.199 + 0.613i)6-s + 4.95·7-s + (0.309 − 0.951i)8-s + (2.08 − 1.51i)9-s + (−0.276 − 2.21i)10-s + (−0.731 − 0.531i)11-s + (0.522 − 0.379i)12-s + (−0.476 + 0.346i)13-s + (−4.00 − 2.91i)14-s + (0.697 − 1.26i)15-s + (−0.809 + 0.587i)16-s + (0.623 − 1.91i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.115 − 0.354i)3-s + (0.154 + 0.475i)4-s + (0.683 + 0.730i)5-s + (−0.0814 + 0.250i)6-s + 1.87·7-s + (0.109 − 0.336i)8-s + (0.696 − 0.506i)9-s + (−0.0873 − 0.701i)10-s + (−0.220 − 0.160i)11-s + (0.150 − 0.109i)12-s + (−0.132 + 0.0959i)13-s + (−1.07 − 0.777i)14-s + (0.180 − 0.326i)15-s + (−0.202 + 0.146i)16-s + (0.151 − 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.63340 - 0.422562i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.63340 - 0.422562i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (-1.52 - 1.63i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
| good | 3 | \( 1 + (0.199 + 0.613i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 4.95T + 7T^{2} \) |
| 11 | \( 1 + (0.731 + 0.531i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (0.476 - 0.346i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.623 + 1.91i)T + (-13.7 - 9.99i)T^{2} \) |
| 23 | \( 1 + (-4.69 - 3.40i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.49 + 4.60i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (2.05 - 6.32i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (6.28 - 4.56i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-3.74 + 2.71i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + (2.47 + 7.62i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.744 - 2.29i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-2.10 + 1.52i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-12.4 - 9.01i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (3.95 - 12.1i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (2.45 + 7.55i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (5.58 + 4.05i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (4.97 + 15.3i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.85 + 11.8i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-5.77 - 4.19i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.95 - 15.2i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.13924773294950947661960717588, −9.160984932937396018405502288571, −8.365979265856265640449005045324, −7.32505248533959476052006726633, −6.98255128937573570090282087479, −5.60620845962186051654221228410, −4.73621696237957502952730967047, −3.37138949702945823174139233764, −2.07486474412662579597310200429, −1.29044693233834602258018199801,
1.32019766631094611003018483248, 2.10600643118833374493016464670, 4.25216967853485452393399991117, 5.07290869058740825560325774863, 5.47124061406480573306875510474, 6.86916230494516899116317421606, 7.84122523911899461635897987874, 8.359061534213586211706162036943, 9.210720522148832893125888882849, 10.07031283727811333202355659575
