| L(s) = 1 | − 3-s − 0.133i·5-s − 3.92i·7-s + 9-s − 5.05i·11-s + 0.133i·15-s + 4.92·17-s − 6.79i·19-s + 3.92i·21-s + 5.05·23-s + 4.98·25-s − 27-s − 3.86·29-s − 1.13i·31-s + 5.05i·33-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.0596i·5-s − 1.48i·7-s + 0.333·9-s − 1.52i·11-s + 0.0344i·15-s + 1.19·17-s − 1.55i·19-s + 0.856i·21-s + 1.05·23-s + 0.996·25-s − 0.192·27-s − 0.717·29-s − 0.203i·31-s + 0.880i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.583924542\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.583924542\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 0.133iT - 5T^{2} \) |
| 7 | \( 1 + 3.92iT - 7T^{2} \) |
| 11 | \( 1 + 5.05iT - 11T^{2} \) |
| 17 | \( 1 - 4.92T + 17T^{2} \) |
| 19 | \( 1 + 6.79iT - 19T^{2} \) |
| 23 | \( 1 - 5.05T + 23T^{2} \) |
| 29 | \( 1 + 3.86T + 29T^{2} \) |
| 31 | \( 1 + 1.13iT - 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 5.19iT - 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 9.05iT - 47T^{2} \) |
| 53 | \( 1 + 1.59T + 53T^{2} \) |
| 59 | \( 1 - 11.8iT - 59T^{2} \) |
| 61 | \( 1 - 5.79T + 61T^{2} \) |
| 67 | \( 1 + 4.07iT - 67T^{2} \) |
| 71 | \( 1 - 1.05iT - 71T^{2} \) |
| 73 | \( 1 + 3.26iT - 73T^{2} \) |
| 79 | \( 1 + 7.24T + 79T^{2} \) |
| 83 | \( 1 - 11.3iT - 83T^{2} \) |
| 89 | \( 1 - 7.73iT - 89T^{2} \) |
| 97 | \( 1 + 7.13iT - 97T^{2} \) |
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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
−8.155417482253507450790167902305, −7.14066557504320839070050961901, −6.98937432247657635553001564806, −5.89203902932904440923910437240, −5.25135489671039050860628107539, −4.41718084189637908177034038363, −3.57299693726454545594161904703, −2.83295805592105250941081257566, −1.06959503081521705052006627166, −0.62620311924091385288034766873,
1.32565043292386332652646597555, 2.23683107475823153177732685682, 3.18914911207464435883712436734, 4.27831096655621500687976491923, 5.14714341016852397960918790408, 5.65570161807973704595896657747, 6.33590770955453641046748516841, 7.30401555952284570577554179246, 7.80413515093878082359758919907, 8.785048056868148268847562046207
