| L(s) = 1 | + 5-s + 7-s − 3·11-s − 5.29·13-s + 5.29·17-s − 5.29·19-s + 5.29·23-s − 4·25-s − 6·29-s + 7·31-s + 35-s + 5.29·37-s + 5.29·41-s + 10.5·43-s − 6·49-s − 9·53-s − 3·55-s + 4·59-s − 5.29·65-s − 15.8·71-s − 3·73-s − 3·77-s − 4·79-s − 7·83-s + 5.29·85-s − 10.5·89-s − 5.29·91-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s − 0.904·11-s − 1.46·13-s + 1.28·17-s − 1.21·19-s + 1.10·23-s − 0.800·25-s − 1.11·29-s + 1.25·31-s + 0.169·35-s + 0.869·37-s + 0.826·41-s + 1.61·43-s − 0.857·49-s − 1.23·53-s − 0.404·55-s + 0.520·59-s − 0.656·65-s − 1.88·71-s − 0.351·73-s − 0.341·77-s − 0.450·79-s − 0.768·83-s + 0.573·85-s − 1.12·89-s − 0.554·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 5.29T + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 7T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 - 5.29T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 9T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 15.8T + 71T^{2} \) |
| 73 | \( 1 + 3T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 7T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 - 7T + 97T^{2} \) |
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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
−7.73413441222485371242283781596, −7.03969701289369536967084359008, −6.04735425783749049069731749705, −5.51475073687053628314313619279, −4.79377556584879184815922596476, −4.14298275975194974020374309816, −2.87065170161267361374176511486, −2.43510499683733921294458785493, −1.35050523037249265804119449307, 0,
1.35050523037249265804119449307, 2.43510499683733921294458785493, 2.87065170161267361374176511486, 4.14298275975194974020374309816, 4.79377556584879184815922596476, 5.51475073687053628314313619279, 6.04735425783749049069731749705, 7.03969701289369536967084359008, 7.73413441222485371242283781596
