| L(s) = 1 | − 2.35·3-s + 2.56·5-s + 4.15i·7-s + 2.56·9-s + 2.33i·11-s + 4.29i·13-s − 6.04·15-s − 17-s + (−3.68 − 2.33i)19-s − 9.79i·21-s − 1.82i·23-s + 1.56·25-s + 1.03·27-s + 1.20i·29-s − 1.32·31-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 1.14·5-s + 1.56i·7-s + 0.853·9-s + 0.703i·11-s + 1.19i·13-s − 1.55·15-s − 0.242·17-s + (−0.844 − 0.535i)19-s − 2.13i·21-s − 0.379i·23-s + 0.312·25-s + 0.198·27-s + 0.223i·29-s − 0.237·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.844 - 0.535i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7470193398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7470193398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (3.68 + 2.33i)T \) |
| good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 - 4.15iT - 7T^{2} \) |
| 11 | \( 1 - 2.33iT - 11T^{2} \) |
| 13 | \( 1 - 4.29iT - 13T^{2} \) |
| 17 | \( 1 + T + 17T^{2} \) |
| 23 | \( 1 + 1.82iT - 23T^{2} \) |
| 29 | \( 1 - 1.20iT - 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 + 5.49iT - 37T^{2} \) |
| 41 | \( 1 + 5.49iT - 41T^{2} \) |
| 43 | \( 1 + 1.30iT - 43T^{2} \) |
| 47 | \( 1 - 6.99iT - 47T^{2} \) |
| 53 | \( 1 - 9.79iT - 53T^{2} \) |
| 59 | \( 1 + 6.33T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 - 0.290T + 67T^{2} \) |
| 71 | \( 1 - 2.06T + 71T^{2} \) |
| 73 | \( 1 + 0.123T + 73T^{2} \) |
| 79 | \( 1 + 13.4T + 79T^{2} \) |
| 83 | \( 1 + 11.9iT - 83T^{2} \) |
| 89 | \( 1 - 14.0iT - 89T^{2} \) |
| 97 | \( 1 - 2.41iT - 97T^{2} \) |
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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
−10.13346505932965829194924797960, −9.074063374172853770410788903704, −8.968707626827045052702916154550, −7.31217727263245740332433123279, −6.25805383625151942455209160977, −6.07165129904256364124301589247, −5.14079191344998227406838762423, −4.44575738756450146125143018283, −2.50888245606522198657795128554, −1.77250422970883646264766990988,
0.37599194479412328489182952397, 1.51556262701591832727803067715, 3.21718395778141571250310120246, 4.40602227316094650033305670253, 5.33044582128154813133030455185, 6.05010660373898966844526961538, 6.59455593105475762006348282665, 7.60791569358433734416185114470, 8.538201240945928620214282917165, 9.930669405528168404247684300554
