| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 11-s − 13-s − 2·14-s + 16-s + 4·17-s − 2·19-s + 20-s + 22-s + 6·23-s + 25-s + 26-s + 2·28-s + 4·29-s − 4·31-s − 32-s − 4·34-s + 2·35-s − 6·37-s + 2·38-s − 40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.970·17-s − 0.458·19-s + 0.223·20-s + 0.213·22-s + 1.25·23-s + 1/5·25-s + 0.196·26-s + 0.377·28-s + 0.742·29-s − 0.718·31-s − 0.176·32-s − 0.685·34-s + 0.338·35-s − 0.986·37-s + 0.324·38-s − 0.158·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.75407818988737, −16.14972463529485, −15.30018880552381, −15.03867487671892, −14.32716163016608, −13.86291461790541, −13.14836337829568, −12.38539016381716, −12.09829744653046, −11.11825753848827, −10.89167191059216, −10.14825957942059, −9.710070468423200, −8.897258483513326, −8.525557190167926, −7.745247407863826, −7.319803607184424, −6.536689981069450, −5.893024667838795, −5.033429411117244, −4.717033965971307, −3.406506194747321, −2.864844221203500, −1.799947701466223, −1.314315028062276, 0,
1.314315028062276, 1.799947701466223, 2.864844221203500, 3.406506194747321, 4.717033965971307, 5.033429411117244, 5.893024667838795, 6.536689981069450, 7.319803607184424, 7.745247407863826, 8.525557190167926, 8.897258483513326, 9.710070468423200, 10.14825957942059, 10.89167191059216, 11.11825753848827, 12.09829744653046, 12.38539016381716, 13.14836337829568, 13.86291461790541, 14.32716163016608, 15.03867487671892, 15.30018880552381, 16.14972463529485, 16.75407818988737
