L(s) = 1 | + (−0.294 + 1.38i)2-s + (0.104 − 0.994i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)5-s + (1.34 + 0.437i)6-s + (1.40 − 0.147i)7-s + (−0.978 − 0.207i)9-s + 1.41i·10-s + (−0.499 + 0.866i)12-s + (−0.209 + 1.98i)14-s + (−0.104 − 0.994i)15-s + (−0.669 − 0.743i)16-s + (0.575 − 1.29i)18-s + (−0.978 − 0.207i)20-s − 1.41i·21-s + ⋯ |
L(s) = 1 | + (−0.294 + 1.38i)2-s + (0.104 − 0.994i)3-s + (−0.913 − 0.406i)4-s + (0.978 − 0.207i)5-s + (1.34 + 0.437i)6-s + (1.40 − 0.147i)7-s + (−0.978 − 0.207i)9-s + 1.41i·10-s + (−0.499 + 0.866i)12-s + (−0.209 + 1.98i)14-s + (−0.104 − 0.994i)15-s + (−0.669 − 0.743i)16-s + (0.575 − 1.29i)18-s + (−0.978 − 0.207i)20-s − 1.41i·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.736 - 0.676i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.158160485\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.158160485\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.104 + 0.994i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.294 - 1.38i)T + (-0.913 - 0.406i)T^{2} \) |
| 5 | \( 1 + (-0.978 + 0.207i)T + (0.913 - 0.406i)T^{2} \) |
| 7 | \( 1 + (-1.40 + 0.147i)T + (0.978 - 0.207i)T^{2} \) |
| 13 | \( 1 + (0.104 + 0.994i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (1.40 - 0.147i)T + (0.978 - 0.207i)T^{2} \) |
| 31 | \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.913 - 0.406i)T + (0.669 - 0.743i)T^{2} \) |
| 53 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \) |
| 61 | \( 1 + (1.05 - 0.946i)T + (0.104 - 0.994i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.831 + 1.14i)T + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 83 | \( 1 + (1.05 - 0.946i)T + (0.104 - 0.994i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.978 + 0.207i)T + (0.913 + 0.406i)T^{2} \) |
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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
−9.780913938632627789123307555218, −8.954839888189361746274498112162, −8.273070012858916757063266056548, −7.60948092129893710968710155010, −7.01153041783487104103839633501, −5.83006522259742373040152351296, −5.65487009880439745244214764988, −4.47850130754752118890978442312, −2.50810393779430471344256421674, −1.48115754420001672506496018547,
1.65494580223565879473431541364, 2.45605676292246374281323341074, 3.54415247109377672229603464551, 4.58717201815765784123149877818, 5.37349177261154745929041675063, 6.39810963985813889872544125275, 7.936408522708452807586447006549, 8.713233266948904268790719432931, 9.496785860998779115992817420606, 10.05261725663487092780729080077