| L(s) = 1 | + (1 + 2i)5-s − 2i·7-s − 5·11-s + 4i·13-s − 6i·17-s + 7·19-s − 4i·23-s + (−3 + 4i)25-s + 5·29-s − 3·31-s + (4 − 2i)35-s + 2i·37-s + 7·41-s + 6i·43-s − 6i·47-s + ⋯ |
| L(s) = 1 | + (0.447 + 0.894i)5-s − 0.755i·7-s − 1.50·11-s + 1.10i·13-s − 1.45i·17-s + 1.60·19-s − 0.834i·23-s + (−0.600 + 0.800i)25-s + 0.928·29-s − 0.538·31-s + (0.676 − 0.338i)35-s + 0.328i·37-s + 1.09·41-s + 0.914i·43-s − 0.875i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.847401355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.847401355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 5T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 - 6iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 15T + 59T^{2} \) |
| 61 | \( 1 - 14T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 5T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 14iT - 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−8.743536939271235925691531680220, −7.68783460011982158259623737201, −7.21706996927050284671217656385, −6.68552202568527071563045213833, −5.59906970894942676522417154246, −4.97174177552502927701994610221, −3.99992380927918166722556576673, −2.88684673687198602138846424768, −2.41147907096045077366422783657, −0.881200048699571666827190425284,
0.75444786489756657512384118261, 1.99822011138683783004596458965, 2.87432077235443808160423520517, 3.86091550604446178400907312555, 5.20287668232690791959495844249, 5.38675684065261324655205415692, 5.99845363328289636055293638855, 7.28757851154520688005287227432, 8.100192861357411231462897940024, 8.402061150344854691751697261604
