| L(s) = 1 | + (−1 − 1.41i)3-s − 3.41·5-s + 3.41·7-s + (−1.00 + 2.82i)9-s − 0.828i·11-s + (3 − 2i)13-s + (3.41 + 4.82i)15-s − 2.82i·17-s − 2.24·19-s + (−3.41 − 4.82i)21-s + 3.65·23-s + 6.65·25-s + (5.00 − 1.41i)27-s − 6.82i·29-s − 9.07·31-s + ⋯ |
| L(s) = 1 | + (−0.577 − 0.816i)3-s − 1.52·5-s + 1.29·7-s + (−0.333 + 0.942i)9-s − 0.249i·11-s + (0.832 − 0.554i)13-s + (0.881 + 1.24i)15-s − 0.685i·17-s − 0.514·19-s + (−0.745 − 1.05i)21-s + 0.762·23-s + 1.33·25-s + (0.962 − 0.272i)27-s − 1.26i·29-s − 1.62·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1248 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6676162558\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6676162558\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1 + 1.41i)T \) |
| 13 | \( 1 + (-3 + 2i)T \) |
| good | 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 + 0.828iT - 11T^{2} \) |
| 17 | \( 1 + 2.82iT - 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 6.82iT - 29T^{2} \) |
| 31 | \( 1 + 9.07T + 31T^{2} \) |
| 37 | \( 1 + 1.65iT - 37T^{2} \) |
| 41 | \( 1 + 3.41T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 - 3.17iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 3.17iT - 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 + 12.5T + 67T^{2} \) |
| 71 | \( 1 + 8.82iT - 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 12.4iT - 79T^{2} \) |
| 83 | \( 1 - 8.82iT - 83T^{2} \) |
| 89 | \( 1 + 11.4T + 89T^{2} \) |
| 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
−9.038991463411923102229260081793, −8.140255575771365986378362654596, −7.83080570476044991189807594123, −7.09410789390747186601858268127, −6.01841265603733401399244164886, −5.04468517592485000555462171185, −4.28905763790457264791401747501, −3.09751758663445219660593593015, −1.59647302647266000234673884498, −0.33449030984361137159076865561,
1.45128650836431266308074633265, 3.38990137390120777782750934147, 4.13276606701004873527094801372, 4.74186103749307424443154411402, 5.66166442523399324393205179612, 6.87232700327780998496989578258, 7.61434321247419915291869442334, 8.702024474735192035383736455541, 8.858536125022809072975036852763, 10.39574375084192473758967170595
