L(s) = 1 | + (0.396 + 0.918i)2-s + (0.893 + 0.448i)3-s + (−0.686 + 0.727i)4-s + (−0.0581 + 0.998i)6-s + (−0.939 − 0.342i)8-s + (0.597 + 0.802i)9-s + (1.14 − 0.754i)11-s + (−0.939 + 0.342i)12-s + (−0.0581 − 0.998i)16-s + (−1.52 − 1.27i)17-s + (−0.500 + 0.866i)18-s + (−1.28 + 1.07i)19-s + (1.14 + 0.754i)22-s + (−0.686 − 0.727i)24-s + (−0.993 − 0.116i)25-s + ⋯ |
L(s) = 1 | + (0.396 + 0.918i)2-s + (0.893 + 0.448i)3-s + (−0.686 + 0.727i)4-s + (−0.0581 + 0.998i)6-s + (−0.939 − 0.342i)8-s + (0.597 + 0.802i)9-s + (1.14 − 0.754i)11-s + (−0.939 + 0.342i)12-s + (−0.0581 − 0.998i)16-s + (−1.52 − 1.27i)17-s + (−0.500 + 0.866i)18-s + (−1.28 + 1.07i)19-s + (1.14 + 0.754i)22-s + (−0.686 − 0.727i)24-s + (−0.993 − 0.116i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.135 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.358444698\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.358444698\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.396 - 0.918i)T \) |
| 3 | \( 1 + (-0.893 - 0.448i)T \) |
good | 5 | \( 1 + (0.993 + 0.116i)T^{2} \) |
| 7 | \( 1 + (-0.893 + 0.448i)T^{2} \) |
| 11 | \( 1 + (-1.14 + 0.754i)T + (0.396 - 0.918i)T^{2} \) |
| 13 | \( 1 + (-0.973 + 0.230i)T^{2} \) |
| 17 | \( 1 + (1.52 + 1.27i)T + (0.173 + 0.984i)T^{2} \) |
| 19 | \( 1 + (1.28 - 1.07i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.893 - 0.448i)T^{2} \) |
| 29 | \( 1 + (0.286 + 0.957i)T^{2} \) |
| 31 | \( 1 + (0.835 - 0.549i)T^{2} \) |
| 37 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 41 | \( 1 + (-0.707 + 1.64i)T + (-0.686 - 0.727i)T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.355i)T + (0.597 + 0.802i)T^{2} \) |
| 47 | \( 1 + (0.835 + 0.549i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.0971 - 0.0639i)T + (0.396 + 0.918i)T^{2} \) |
| 61 | \( 1 + (0.0581 - 0.998i)T^{2} \) |
| 67 | \( 1 + (-1.16 - 1.56i)T + (-0.286 + 0.957i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (1.12 + 0.408i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (0.686 - 0.727i)T^{2} \) |
| 83 | \( 1 + (-0.137 - 0.318i)T + (-0.686 + 0.727i)T^{2} \) |
| 89 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.0333 - 0.572i)T + (-0.993 + 0.116i)T^{2} \) |
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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
−10.97793103197784265593241307545, −9.789076582628220501039432000450, −8.927902064064333592374109322316, −8.553845213891898032951248627334, −7.46647380153620511062042786542, −6.61719362663302468043039272866, −5.62528201469657750610477159018, −4.29391630568579765366176441828, −3.84330283078782636166832966621, −2.43172516855255334981062552482,
1.67990703480025224177193494747, 2.52149860568947923448255700738, 4.00511425290022442036443309756, 4.39069880315795297957636059758, 6.17880278888877936853554286166, 6.81923203993889202170599312770, 8.206066929899026227477642147435, 9.017472445890334369896893697320, 9.531090666022575825633733501488, 10.63116849617570988261757878145