L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.266 − 1.50i)3-s + (0.766 − 0.642i)4-s + (0.266 + 1.50i)6-s + (−0.500 + 0.866i)8-s + (−1.26 − 0.460i)9-s + (0.939 − 1.62i)11-s + (−0.766 − 1.32i)12-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + 1.34·18-s + (−0.326 + 1.85i)22-s + (1.17 + 0.984i)24-s + (0.173 + 0.984i)25-s + (−0.266 + 0.460i)27-s + ⋯ |
L(s) = 1 | + (−0.939 + 0.342i)2-s + (0.266 − 1.50i)3-s + (0.766 − 0.642i)4-s + (0.266 + 1.50i)6-s + (−0.500 + 0.866i)8-s + (−1.26 − 0.460i)9-s + (0.939 − 1.62i)11-s + (−0.766 − 1.32i)12-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s + 1.34·18-s + (−0.326 + 1.85i)22-s + (1.17 + 0.984i)24-s + (0.173 + 0.984i)25-s + (−0.266 + 0.460i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9263767713\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9263767713\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 43 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 97 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
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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
−8.544124621465343456033889577869, −8.011217023921819170900712974774, −7.24148987946682066226870050408, −6.69929505224733914017651950856, −5.97160083327572019165049078227, −5.36156064732578856847028743012, −3.56306633726354533904744297740, −2.73123077938146838251121187694, −1.55461189943639786045468567311, −0.849039973070957189240485347789,
1.55290188225860965339383496743, 2.67884977988730874826374412182, 3.66733435266661490581066128895, 4.25174743023063751662454771625, 5.10222619559530843675212547993, 6.30714867613046697952901626406, 7.07336821073377879007933534510, 7.969095340289917000045305495235, 8.687518145256356954384629001254, 9.381684380466953954773262847091