L(s) = 1 | − i·2-s − 1.73·3-s − 4-s + 1.73i·6-s + (2 − 1.73i)7-s + i·8-s + 2.99·9-s − i·11-s + 1.73·12-s + (−1.73 − 2i)14-s + 16-s − 3.46·17-s − 2.99i·18-s − 3.46i·19-s + (−3.46 + 2.99i)21-s − 22-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 1.00·3-s − 0.5·4-s + 0.707i·6-s + (0.755 − 0.654i)7-s + 0.353i·8-s + 0.999·9-s − 0.301i·11-s + 0.500·12-s + (−0.462 − 0.534i)14-s + 0.250·16-s − 0.840·17-s − 0.707i·18-s − 0.794i·19-s + (−0.755 + 0.654i)21-s − 0.213·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.287477 - 0.771079i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.287477 - 0.771079i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + 1.73T \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| 11 | \( 1 + iT \) |
good | 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 10.3T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 - 6.92iT - 97T^{2} \) |
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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
−10.79316633285753591527528346298, −10.21437810286345613394230379287, −9.089520992935260354737427499249, −7.987607169942069840158269168813, −6.94540834172087397037375155189, −5.85855308463100640542285659787, −4.69999864798053547169839904242, −4.04976401376089661775401427431, −2.17769200074351655647976243777, −0.59658248094432583186866255363,
1.72227542664204167521647047351, 3.91331429197674510898519512920, 5.07782702454631711925128064231, 5.64691619956599129925532694512, 6.71938939171693366053646255990, 7.58303265437170784155467966210, 8.587946822976677300145062351869, 9.580909861396310129045780278911, 10.53912245175360079577945345366, 11.49295346764607658133851495016