| L(s) = 1 | + (1.58 + 0.707i)3-s + 2.23i·5-s + 3.16·7-s + (2.00 + 2.23i)9-s + (−1.58 + 3.53i)15-s + (5.00 + 2.23i)21-s + 1.41i·23-s − 5.00·25-s + (1.58 + 4.94i)27-s − 8.94i·29-s + 7.07i·35-s − 4.47i·41-s − 3.16·43-s + (−5.00 + 4.47i)45-s + 9.89i·47-s + ⋯ |
| L(s) = 1 | + (0.912 + 0.408i)3-s + 0.999i·5-s + 1.19·7-s + (0.666 + 0.745i)9-s + (−0.408 + 0.912i)15-s + (1.09 + 0.487i)21-s + 0.294i·23-s − 1.00·25-s + (0.304 + 0.952i)27-s − 1.66i·29-s + 1.19i·35-s − 0.698i·41-s − 0.482·43-s + (−0.745 + 0.666i)45-s + 1.44i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.06821 + 1.34068i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.06821 + 1.34068i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.58 - 0.707i)T \) |
| 5 | \( 1 - 2.23iT \) |
| good | 7 | \( 1 - 3.16T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 - 1.41iT - 23T^{2} \) |
| 29 | \( 1 + 8.94iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 4.47iT - 41T^{2} \) |
| 43 | \( 1 + 3.16T + 43T^{2} \) |
| 47 | \( 1 - 9.89iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 15.8T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 - 15.5iT - 83T^{2} \) |
| 89 | \( 1 + 17.8iT - 89T^{2} \) |
| 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
−10.09918147156470916394888339701, −9.447494592976762206522898684300, −8.339036364232423825051167161626, −7.82349130231537170379600581113, −7.03797302903605784490431539283, −5.84857104336744611462568660556, −4.70210514316095690823358555231, −3.85524527093093490983351278175, −2.75718612746241077752249022575, −1.81563388820170432072503923229,
1.18057131215898341326136873701, 2.09237432031472218882358090108, 3.51892395291180246247410340747, 4.59109289122655203749859738765, 5.30643650543077075715843712203, 6.63659684924697715024659837508, 7.61990677152693008320141212830, 8.308523863162257949302961168234, 8.808130837829987793388007878989, 9.630917131594831564579865412902
