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
−9.964144387205616760757576587111, −8.829965770867876428337453515507, −8.476345920140551803579788351928, −7.77071468299319691845252194904, −6.18426617767866734472035526609, −5.77843658959272649656695787938, −4.91613989710912956000428964026, −3.66397785011834080852943130608, −2.91412062726369648038873762917, −1.61463216684733902763759458720,
0.52085885221818334410213551925, 2.18329792223318467754764112735, 2.90866236358056530313661970435, 4.30562658207975681101140838961, 4.96146644002096163886826022258, 6.40802998149937015491503655251, 6.94406259940964775861221891901, 7.57881509584073805797410453542, 8.437734929599080839022445815197, 9.472059038176369575204979014707