L(s) = 1 | + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)9-s + (1.43 + 1.20i)11-s + (−0.173 − 0.300i)17-s + (0.766 − 1.32i)19-s + (0.173 − 0.984i)25-s + (0.500 − 0.866i)27-s + (−0.326 − 1.85i)33-s + (0.266 + 1.50i)41-s + (−0.266 − 0.223i)43-s + (0.766 − 0.642i)49-s + (−0.0603 + 0.342i)51-s + (−1.43 + 0.524i)57-s + (−1.17 + 0.984i)59-s + (0.326 + 1.85i)67-s + ⋯ |
L(s) = 1 | + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)9-s + (1.43 + 1.20i)11-s + (−0.173 − 0.300i)17-s + (0.766 − 1.32i)19-s + (0.173 − 0.984i)25-s + (0.500 − 0.866i)27-s + (−0.326 − 1.85i)33-s + (0.266 + 1.50i)41-s + (−0.266 − 0.223i)43-s + (0.766 − 0.642i)49-s + (−0.0603 + 0.342i)51-s + (−1.43 + 0.524i)57-s + (−1.17 + 0.984i)59-s + (0.326 + 1.85i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8430595438\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8430595438\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.766 + 0.642i)T \) |
good | 5 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 11 | \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 13 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \) |
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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
−10.32603388636591074661355910632, −9.538086466294714446199218127048, −8.665370753904041975799285980307, −7.43566986774318714601768681980, −6.90469640220417199107341311042, −6.15767982291312986881143777178, −4.96465952706016470716110972001, −4.25728877682293028888567227115, −2.58004789456972379620689128136, −1.27439171901337151242175599893,
1.30184874323710865942771920751, 3.41635591593603092218720761330, 3.97024591520817943054651417412, 5.27353005254842872102181965405, 6.00479164894248623330059057604, 6.72904052626321690493976237270, 7.934542031871418280105880249076, 9.048590610722006175178668135896, 9.483789513685366094211440820451, 10.60217938060682259833517069392