L(s) = 1 | + (0.642 + 0.766i)2-s + (−0.524 + 1.43i)3-s + (−0.173 + 0.984i)4-s + (−1.43 + 0.524i)6-s + (−0.866 + 0.500i)8-s + (−1.03 − 0.866i)9-s + (0.939 + 1.62i)11-s + (−1.32 − 0.766i)12-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s − 1.34i·18-s + (−0.173 + 0.984i)19-s + (−0.642 + 1.76i)22-s + (−0.266 − 1.50i)24-s + (0.460 − 0.266i)27-s + ⋯ |
L(s) = 1 | + (0.642 + 0.766i)2-s + (−0.524 + 1.43i)3-s + (−0.173 + 0.984i)4-s + (−1.43 + 0.524i)6-s + (−0.866 + 0.500i)8-s + (−1.03 − 0.866i)9-s + (0.939 + 1.62i)11-s + (−1.32 − 0.766i)12-s + (−0.939 − 0.342i)16-s + (−0.642 − 0.766i)17-s − 1.34i·18-s + (−0.173 + 0.984i)19-s + (−0.642 + 1.76i)22-s + (−0.266 − 1.50i)24-s + (0.460 − 0.266i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.229087059\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.229087059\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.642 - 0.766i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (0.173 - 0.984i)T \) |
good | 3 | \( 1 + (0.524 - 1.43i)T + (-0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 17 | \( 1 + (0.642 + 0.766i)T + (-0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (0.326 + 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 43 | \( 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2} \) |
| 47 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 59 | \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \) |
| 61 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 67 | \( 1 + (0.223 - 0.266i)T + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (-0.642 + 1.76i)T + (-0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.300 - 0.173i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.223 - 0.266i)T + (-0.173 + 0.984i)T^{2} \) |
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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
−9.296195182517455550458953697917, −8.459730857418184125065479628745, −7.48442354560821613501645261873, −6.78042704776828942516978160864, −6.10939846765994611163540489138, −5.16906595142857604713880411085, −4.68260760100712480164423913007, −4.07943053464766651834825326893, −3.41671489553824395507744801762, −2.09458520194983912617857706974,
0.60888182866784265040424091979, 1.51050298676087786511091072176, 2.43548109833514510922437320191, 3.41882111367050570248803044137, 4.26858406646877009897720155347, 5.40952627679982851534070383057, 5.96769643401102341492227937276, 6.62906988442498469553413014394, 7.06623866303850991088800936899, 8.474686016589772901744160567884