L(s) = 1 | + (−0.766 + 0.642i)3-s + (1.11 + 0.642i)7-s + (0.173 − 0.984i)9-s + (0.439 − 0.524i)13-s + (−0.5 + 0.866i)19-s + (−1.26 + 0.223i)21-s + (0.766 + 0.642i)25-s + (0.500 + 0.866i)27-s + (−0.173 + 0.300i)31-s + 1.96i·37-s + 0.684i·39-s + (0.673 − 1.85i)43-s + (0.326 + 0.565i)49-s + (−0.173 − 0.984i)57-s + (−1.76 + 0.642i)61-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)3-s + (1.11 + 0.642i)7-s + (0.173 − 0.984i)9-s + (0.439 − 0.524i)13-s + (−0.5 + 0.866i)19-s + (−1.26 + 0.223i)21-s + (0.766 + 0.642i)25-s + (0.500 + 0.866i)27-s + (−0.173 + 0.300i)31-s + 1.96i·37-s + 0.684i·39-s + (0.673 − 1.85i)43-s + (0.326 + 0.565i)49-s + (−0.173 − 0.984i)57-s + (−1.76 + 0.642i)61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.624 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8865134173\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8865134173\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.766 - 0.642i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (-1.11 - 0.642i)T + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.439 + 0.524i)T + (-0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 1.96iT - T^{2} \) |
| 41 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.673 + 1.85i)T + (-0.766 - 0.642i)T^{2} \) |
| 47 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 59 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (1.76 - 0.642i)T + (0.766 - 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (1.70 - 0.300i)T + (0.939 - 0.342i)T^{2} \) |
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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
−10.67921681640189324714043900808, −9.664023848962462170502391955582, −8.726701956571249256715200532494, −8.088580974257202841202410105078, −6.87450484162422450138840103414, −5.85187099726469081950552789103, −5.21575644356347438825486665760, −4.36106983227531909691483632594, −3.19907368848410111279421366728, −1.55189613574610043673277534186,
1.15678897074856065061015346706, 2.37304353799452129710216815034, 4.19085565706533795696502256892, 4.84743192540101352413871931320, 5.92167410219627062952952555269, 6.80407470476834646452168645921, 7.55588288845736391826192162330, 8.314366657822335972157386743959, 9.290937859256195661131792514391, 10.58372721198729268980463675052