L(s) = 1 | − 2.45·2-s − 1.73i·3-s + 4.03·4-s − 2.23i·5-s + 4.25i·6-s − 4.98·8-s − 2.99·9-s + 5.49i·10-s − 6.98i·12-s − 3.87·15-s + 4.19·16-s + 8.06i·17-s + 7.36·18-s + 5.62i·19-s − 9.01i·20-s + ⋯ |
L(s) = 1 | − 1.73·2-s − 0.999i·3-s + 2.01·4-s − 0.999i·5-s + 1.73i·6-s − 1.76·8-s − 0.999·9-s + 1.73i·10-s − 2.01i·12-s − 0.999·15-s + 1.04·16-s + 1.95i·17-s + 1.73·18-s + 1.29i·19-s − 2.01i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.105812 + 0.0903608i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.105812 + 0.0903608i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 2.45T + 2T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 8.06iT - 17T^{2} \) |
| 19 | \( 1 - 5.62iT - 19T^{2} \) |
| 23 | \( 1 + 9.58T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 4.42iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 1.02iT - 47T^{2} \) |
| 53 | \( 1 + 9.43T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 4.08iT - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 5.83T + 79T^{2} \) |
| 83 | \( 1 - 15.0iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−10.30581538059066100908738626481, −9.627435645523080844114932537576, −8.549773154927706262170217431630, −8.156229238312469159048113273068, −7.63752760242722177302819760174, −6.29807095568265224447911809010, −5.81895527250229912718068214538, −3.93359762789124869717248049998, −2.05392429950286131786965678221, −1.40289441875517714861860986961,
0.12507660068975628630038972586, 2.32784207992758966728807847293, 3.19156109608678093738325344843, 4.71264394531730523985265015786, 6.07256354926077032015518273081, 6.99435871751352390872490119140, 7.75058163495239748698591363222, 8.715128146893724480153758924128, 9.543453078415377929144170041347, 9.948666060112340194642378461186