L(s) = 1 | + (−0.608 − 0.793i)3-s + (0.5 − 0.866i)4-s + (−0.923 − 1.60i)5-s + (−0.258 + 0.965i)9-s + (−0.866 − 0.5i)11-s + (−0.991 + 0.130i)12-s + (−0.707 + 1.70i)15-s + (−0.499 − 0.866i)16-s − 1.84·20-s + (1.22 − 0.707i)23-s + (−1.20 + 2.09i)25-s + (0.923 − 0.382i)27-s + (0.662 + 0.382i)31-s + (0.130 + 0.991i)33-s + (0.707 + 0.707i)36-s + ⋯ |
L(s) = 1 | + (−0.608 − 0.793i)3-s + (0.5 − 0.866i)4-s + (−0.923 − 1.60i)5-s + (−0.258 + 0.965i)9-s + (−0.866 − 0.5i)11-s + (−0.991 + 0.130i)12-s + (−0.707 + 1.70i)15-s + (−0.499 − 0.866i)16-s − 1.84·20-s + (1.22 − 0.707i)23-s + (−1.20 + 2.09i)25-s + (0.923 − 0.382i)27-s + (0.662 + 0.382i)31-s + (0.130 + 0.991i)33-s + (0.707 + 0.707i)36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6496643569\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6496643569\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.608 + 0.793i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.866 + 0.5i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (-0.382 - 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 - 1.41iT - T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.382 + 0.662i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + 1.84iT - T^{2} \) |
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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
−8.964494669990809270661294979819, −8.279485858511457747352274736441, −7.58969095329151995184010540888, −6.73489083983771808632934428022, −5.76629250942013126623457496955, −5.10292121668770824299558191888, −4.53888456459669180725722749016, −2.86332145453782675770087704702, −1.48980248194149513341674096962, −0.55598256061680394451580002487,
2.55669497375242109014412310694, 3.24928426564777476402100286321, 3.96955614368970561489755654731, 4.95544282495927768222046163048, 6.19097720980376963960514045517, 6.90283612040654989381623024438, 7.51224248007693192224504598426, 8.209359446120310600883824235916, 9.373182884544096350235876936637, 10.32299133266945222572152239552