L(s) = 1 | + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)11-s + 4·13-s + 0.999·15-s + (−2 + 3.46i)19-s + (−2.5 + 0.866i)21-s + (4 − 6.92i)23-s + (2 + 3.46i)25-s − 0.999·27-s − 3·29-s + (−2.5 − 4.33i)31-s + (−1.5 + 2.59i)33-s + ⋯ |
L(s) = 1 | + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.188 + 0.981i)7-s + (−0.166 + 0.288i)9-s + (0.452 + 0.783i)11-s + 1.10·13-s + 0.258·15-s + (−0.458 + 0.794i)19-s + (−0.545 + 0.188i)21-s + (0.834 − 1.44i)23-s + (0.400 + 0.692i)25-s − 0.192·27-s − 0.557·29-s + (−0.449 − 0.777i)31-s + (−0.261 + 0.452i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35293 + 0.670688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35293 + 0.670688i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3 - 5.19i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 7T + 83T^{2} \) |
| 89 | \( 1 + (-9 + 15.5i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−11.68679580528501483916401391011, −10.70482832250080415851759366785, −9.683454707047541994380299294704, −8.913724464164031676128686744716, −8.275549315163550816236887973917, −6.75880415736497526996388806916, −5.73921741878505684235593132581, −4.66742853115751436336700555480, −3.43472352107102871488601814606, −1.92782916575196486636189341960,
1.20686075024325120967931599859, 3.04168786639307196021575513186, 4.04881298692674209535868395267, 5.74372940352540178256403532821, 6.70819660452786775820537604428, 7.46330659107816661056272453218, 8.654119078104937237731057740227, 9.416301749636659985187879958376, 10.87646019263452338569845838805, 11.04435583558840974334112606251