| L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.320 − 0.987i)3-s + (0.309 + 0.951i)4-s + (−0.846 + 2.06i)5-s + (0.320 − 0.987i)6-s − 2.76·7-s + (−0.309 + 0.951i)8-s + (1.55 − 1.12i)9-s + (−1.90 + 1.17i)10-s + (−3.42 − 2.48i)11-s + (0.840 − 0.610i)12-s + (0.00617 − 0.00448i)13-s + (−2.24 − 1.62i)14-s + (2.31 + 0.171i)15-s + (−0.809 + 0.587i)16-s + (−1.77 + 5.47i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.185 − 0.570i)3-s + (0.154 + 0.475i)4-s + (−0.378 + 0.925i)5-s + (0.131 − 0.403i)6-s − 1.04·7-s + (−0.109 + 0.336i)8-s + (0.518 − 0.376i)9-s + (−0.601 + 0.372i)10-s + (−1.03 − 0.749i)11-s + (0.242 − 0.176i)12-s + (0.00171 − 0.00124i)13-s + (−0.598 − 0.434i)14-s + (0.597 + 0.0443i)15-s + (−0.202 + 0.146i)16-s + (−0.431 + 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 + 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0203027 - 0.113284i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0203027 - 0.113284i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.846 - 2.06i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| good | 3 | \( 1 + (0.320 + 0.987i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 + (3.42 + 2.48i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.00617 + 0.00448i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (1.77 - 5.47i)T + (-13.7 - 9.99i)T^{2} \) |
| 23 | \( 1 + (6.12 + 4.45i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (0.440 + 1.35i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.117 + 0.360i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (5.43 - 3.94i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (6.16 - 4.47i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 5.65T + 43T^{2} \) |
| 47 | \( 1 + (1.63 + 5.01i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (3.22 + 9.93i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (-5.81 + 4.22i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-1.09 - 0.797i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (4.48 - 13.8i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-2.61 - 8.04i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-8.92 - 6.48i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (5.17 + 15.9i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (4.38 - 13.4i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (1.13 + 0.825i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.06 - 12.5i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.40784955660376934612406111468, −9.980011088373180625590200677577, −8.429320491027916163121450523931, −7.891765944018154102778577002068, −6.71490618073494299860161978091, −6.49282755937326357913786591609, −5.62542752597389645195926749240, −4.08653046579227644989762811645, −3.39724410501160893849523441645, −2.26518803039329321054607015052,
0.04160601854458574640640115082, 1.98487094753942487953035101770, 3.30092510170632156489538723002, 4.31511845187461397768746914523, 4.97512389801756291146681448987, 5.72980939526973908568420753437, 7.07456022466631176024441333114, 7.75405362387316730251421239677, 9.138741586270851996987211178882, 9.667674758490811371649910612630
