L(s) = 1 | + (0.266 − 0.223i)2-s + (−0.326 − 0.118i)3-s + (−0.152 + 0.866i)4-s + (−0.113 + 0.0412i)6-s + (−0.173 + 0.300i)7-s + (0.326 + 0.565i)8-s + (−0.673 − 0.565i)9-s + (−0.5 − 0.866i)11-s + (0.152 − 0.264i)12-s + (−0.939 + 0.342i)13-s + (0.0209 + 0.118i)14-s + (−0.613 − 0.223i)16-s − 0.305·18-s + (−0.939 + 0.342i)19-s + (0.0923 − 0.0775i)21-s + (−0.326 − 0.118i)22-s + ⋯ |
L(s) = 1 | + (0.266 − 0.223i)2-s + (−0.326 − 0.118i)3-s + (−0.152 + 0.866i)4-s + (−0.113 + 0.0412i)6-s + (−0.173 + 0.300i)7-s + (0.326 + 0.565i)8-s + (−0.673 − 0.565i)9-s + (−0.5 − 0.866i)11-s + (0.152 − 0.264i)12-s + (−0.939 + 0.342i)13-s + (0.0209 + 0.118i)14-s + (−0.613 − 0.223i)16-s − 0.305·18-s + (−0.939 + 0.342i)19-s + (0.0923 − 0.0775i)21-s + (−0.326 − 0.118i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2717 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.09656834671\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.09656834671\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 + (0.939 - 0.342i)T \) |
good | 2 | \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 3 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
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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
−8.594861203181247697785150784745, −8.097045797475651880245351271932, −7.10296381121931181394311411726, −6.36443844226671237887926190447, −5.54026051557747399953151835566, −4.72059698282014735693556079454, −3.78028590591536921524512424383, −2.94843502921775595677434703082, −2.18227609522745234297158898331, −0.05381897653259440125134448370,
1.74684361876924131813740933334, 2.72439391837648921995365950697, 4.08912723164233618191876140214, 4.86863480272926001260602878207, 5.40891265580055645148892032792, 6.12593697874058697704892635024, 7.07720592954123532671735107914, 7.63160413486727805164949300600, 8.616282678853559985581503800210, 9.554723371208068562235374397829