L(s) = 1 | + (−0.939 − 0.342i)2-s + 3-s + (0.766 + 0.642i)4-s + (−0.939 − 0.342i)6-s + (−0.500 − 0.866i)8-s + 9-s + 0.347·11-s + (0.766 + 0.642i)12-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + (−0.939 − 0.342i)18-s + (0.766 + 0.642i)19-s + (−0.326 − 0.118i)22-s + (−0.500 − 0.866i)24-s + (−0.939 + 0.342i)25-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + 3-s + (0.766 + 0.642i)4-s + (−0.939 − 0.342i)6-s + (−0.500 − 0.866i)8-s + 9-s + 0.347·11-s + (0.766 + 0.642i)12-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + (−0.939 − 0.342i)18-s + (0.766 + 0.642i)19-s + (−0.326 − 0.118i)22-s + (−0.500 − 0.866i)24-s + (−0.939 + 0.342i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.057326657\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.057326657\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 - T \) |
| 19 | \( 1 + (-0.766 - 0.642i)T \) |
good | 5 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 - 0.347T + T^{2} \) |
| 13 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 53 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
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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
−9.463964477065985139930029797403, −9.172499390441546929120712464767, −8.194053232253698939089370993575, −7.52894145522254680623676109656, −6.93552878311931629290211801109, −5.75309293635175892666799738823, −4.28381027611245566600458924389, −3.38626113087263582932418115737, −2.48896422121217798282457803573, −1.37812690684402521232657985478,
1.43164168372506385630254904679, 2.48368565352088062021267999577, 3.58658584976803941919363871206, 4.75983082028672499142384832834, 6.01720066092493480435067574121, 6.77606976702368184418140180853, 7.70549995224751697165440002550, 8.162844698632471218921117496591, 9.094481989113774575223253923513, 9.537450023978616118920145028996