L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.73 + 0.564i)5-s − 0.999i·8-s + (−0.913 + 0.406i)9-s + (1.78 + 0.379i)10-s + (0.685 − 1.05i)13-s + (−0.5 + 0.866i)16-s + (0.243 − 1.53i)17-s + (0.994 + 0.104i)18-s + (−1.35 − 1.22i)20-s + (1.89 − 1.37i)25-s + (−1.12 + 0.571i)26-s + (−0.773 − 1.73i)29-s + (0.866 − 0.499i)32-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.73 + 0.564i)5-s − 0.999i·8-s + (−0.913 + 0.406i)9-s + (1.78 + 0.379i)10-s + (0.685 − 1.05i)13-s + (−0.5 + 0.866i)16-s + (0.243 − 1.53i)17-s + (0.994 + 0.104i)18-s + (−1.35 − 1.22i)20-s + (1.89 − 1.37i)25-s + (−1.12 + 0.571i)26-s + (−0.773 − 1.73i)29-s + (0.866 − 0.499i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 964 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.347 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3156354411\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3156354411\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 241 | \( 1 + T \) |
good | 3 | \( 1 + (0.913 - 0.406i)T^{2} \) |
| 5 | \( 1 + (1.73 - 0.564i)T + (0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (0.207 + 0.978i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (-0.685 + 1.05i)T + (-0.406 - 0.913i)T^{2} \) |
| 17 | \( 1 + (-0.243 + 1.53i)T + (-0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (0.773 + 1.73i)T + (-0.669 + 0.743i)T^{2} \) |
| 31 | \( 1 + (-0.207 - 0.978i)T^{2} \) |
| 37 | \( 1 + (0.770 + 1.18i)T + (-0.406 + 0.913i)T^{2} \) |
| 41 | \( 1 + (-1.01 - 1.40i)T + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 47 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.0850i)T + (0.978 + 0.207i)T^{2} \) |
| 59 | \( 1 + (0.913 + 0.406i)T^{2} \) |
| 61 | \( 1 + (1.53 + 0.5i)T + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.978 + 0.207i)T^{2} \) |
| 71 | \( 1 + (0.743 + 0.669i)T^{2} \) |
| 73 | \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.913 - 0.406i)T^{2} \) |
| 89 | \( 1 + (0.434 + 1.62i)T + (-0.866 + 0.5i)T^{2} \) |
| 97 | \( 1 + (-0.104 + 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
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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
−10.09490094547358517592885425556, −9.031414157261053379684966497404, −8.154291735759165357529967660924, −7.75033512011796867114361780441, −7.02539775547543583955386626018, −5.74914722386897586777181880959, −4.30158816435040269424703754421, −3.31739225414411484253127068154, −2.66632131201813639321400064538, −0.42626047175224010861918400919,
1.41319819404841582937990496285, 3.33852614597022256097534378719, 4.25337781287146049617024704511, 5.44314090987887125781755874917, 6.42938203488227925276376858352, 7.31328204841673613917600261937, 8.130207671190807026451152975075, 8.765904945645062786795522964738, 9.143397110384565432206116103638, 10.70760882423743966066006162465