L(s) = 1 | + (−0.5 − 0.866i)3-s + (1 − 1.73i)5-s + (2.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−3 − 5.19i)11-s + 3·13-s − 1.99·15-s + (−2 − 3.46i)17-s + (−2.5 + 4.33i)19-s + (−2 − 1.73i)21-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + 0.999·27-s + 4·29-s + (−3.5 − 6.06i)31-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.447 − 0.774i)5-s + (0.944 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.904 − 1.56i)11-s + 0.832·13-s − 0.516·15-s + (−0.485 − 0.840i)17-s + (−0.573 + 0.993i)19-s + (−0.436 − 0.377i)21-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + 0.192·27-s + 0.742·29-s + (−0.628 − 1.08i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.530569608\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.530569608\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-4 - 6.92i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.5 + 12.9i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.075254067065326305929321751854, −8.366165907148431719707747166390, −7.995824062294730994288961410975, −6.79549591868374692899320467660, −5.85875984115795608758901075346, −5.27045819999909117227030853843, −4.36233391950209015420596156442, −3.01464359590303729352082080501, −1.70312533232172950039168046172, −0.65772956934483450366686268977,
1.78086551505235290708553481262, 2.66294329496635965840391789019, 4.02551398396858222818184726107, 4.90310028084826248753660019944, 5.59408299019937659414643583057, 6.68783967896404805240251239016, 7.30084185744788646178363752566, 8.446274712940039282657322985582, 9.046195217543252511145420893041, 10.18644630176911971846820965572