L(s) = 1 | + (0.809 + 0.587i)2-s + (0.739 + 2.27i)3-s + (0.309 + 0.951i)4-s + (−2.19 − 0.441i)5-s + (−0.739 + 2.27i)6-s − 0.427·7-s + (−0.309 + 0.951i)8-s + (−2.20 + 1.60i)9-s + (−1.51 − 1.64i)10-s + (−4.15 − 3.01i)11-s + (−1.93 + 1.40i)12-s + (−3.25 + 2.36i)13-s + (−0.345 − 0.251i)14-s + (−0.616 − 5.31i)15-s + (−0.809 + 0.587i)16-s + (−0.526 + 1.61i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (0.427 + 1.31i)3-s + (0.154 + 0.475i)4-s + (−0.980 − 0.197i)5-s + (−0.302 + 0.929i)6-s − 0.161·7-s + (−0.109 + 0.336i)8-s + (−0.736 + 0.535i)9-s + (−0.478 − 0.520i)10-s + (−1.25 − 0.909i)11-s + (−0.559 + 0.406i)12-s + (−0.903 + 0.656i)13-s + (−0.0923 − 0.0671i)14-s + (−0.159 − 1.37i)15-s + (−0.202 + 0.146i)16-s + (−0.127 + 0.392i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.303335 - 0.789570i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.303335 - 0.789570i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.19 + 0.441i)T \) |
| 19 | \( 1 + (-0.309 + 0.951i)T \) |
good | 3 | \( 1 + (-0.739 - 2.27i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 0.427T + 7T^{2} \) |
| 11 | \( 1 + (4.15 + 3.01i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (3.25 - 2.36i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.526 - 1.61i)T + (-13.7 - 9.99i)T^{2} \) |
| 23 | \( 1 + (7.71 + 5.60i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.83 - 8.73i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.909 - 2.79i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-2.34 + 1.70i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (7.46 - 5.42i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 + (-3.83 - 11.8i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.671 + 2.06i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (1.15 - 0.841i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.94 + 3.59i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-0.545 + 1.67i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (0.696 + 2.14i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.83 - 1.33i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-1.70 - 5.23i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.722 + 2.22i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (2.25 + 1.63i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (2.41 + 7.44i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.64572304888526963686098481064, −9.707698840474211822168893657344, −8.705663291918266585045746699991, −8.189553212607673111039342755536, −7.27551405141862542124483080837, −6.12330883088195358722914729494, −4.90870018867171317918730862958, −4.50689257733933472036198622428, −3.50527840108314593044309376843, −2.73106778421303302523702854786,
0.28520338306779711560166187589, 2.09485176462322183406265951335, 2.75734297380717353674561387536, 3.98327489612706871849548166033, 5.05544781252197751708182101273, 6.12264293554298314568929913644, 7.34313422341686415096472780741, 7.56703012652629223509899569402, 8.295513663907513467524980799385, 9.785425952471902725830641375406