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 + (−2.5 + 4.33i)11-s + 0.999·15-s + (2 − 3.46i)17-s + (−4 − 6.92i)19-s + (−2.5 − 0.866i)21-s + (−2 − 3.46i)23-s + (2 − 3.46i)25-s − 0.999·27-s + 5·29-s + (1.5 − 2.59i)31-s + (2.5 + 4.33i)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 + (−0.753 + 1.30i)11-s + 0.258·15-s + (0.485 − 0.840i)17-s + (−0.917 − 1.58i)19-s + (−0.545 − 0.188i)21-s + (−0.417 − 0.722i)23-s + (0.400 − 0.692i)25-s − 0.192·27-s + 0.928·29-s + (0.269 − 0.466i)31-s + (0.435 + 0.753i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.354607567\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.354607567\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.5 - 4.33i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4 + 6.92i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2 + 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.5 + 9.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3 + 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1 + 1.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 + (5 - 8.66i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.5 - 2.59i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 7T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−9.472059038176369575204979014707, −8.437734929599080839022445815197, −7.57881509584073805797410453542, −6.94406259940964775861221891901, −6.40802998149937015491503655251, −4.96146644002096163886826022258, −4.30562658207975681101140838961, −2.90866236358056530313661970435, −2.18329792223318467754764112735, −0.52085885221818334410213551925,
1.61463216684733902763759458720, 2.91412062726369648038873762917, 3.66397785011834080852943130608, 4.91613989710912956000428964026, 5.77843658959272649656695787938, 6.18426617767866734472035526609, 7.77071468299319691845252194904, 8.476345920140551803579788351928, 8.829965770867876428337453515507, 9.964144387205616760757576587111