| L(s) = 1 | + 2·5-s − 11-s + 6·13-s + 6·17-s − 8·19-s − 25-s − 6·29-s − 6·37-s − 10·41-s + 8·43-s + 6·53-s − 2·55-s − 4·59-s − 2·61-s + 12·65-s + 12·67-s + 8·71-s − 2·73-s − 4·79-s + 12·83-s + 12·85-s − 6·89-s − 16·95-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.301·11-s + 1.66·13-s + 1.45·17-s − 1.83·19-s − 1/5·25-s − 1.11·29-s − 0.986·37-s − 1.56·41-s + 1.21·43-s + 0.824·53-s − 0.269·55-s − 0.520·59-s − 0.256·61-s + 1.48·65-s + 1.46·67-s + 0.949·71-s − 0.234·73-s − 0.450·79-s + 1.31·83-s + 1.30·85-s − 0.635·89-s − 1.64·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.87831275644477, −12.58147860193869, −12.01833102264666, −11.44342568641546, −10.96301871306526, −10.55267354615699, −10.22748038497542, −9.749882885142702, −9.153484649892942, −8.776449580161769, −8.314685175031064, −7.891072036909860, −7.332403378633052, −6.585976413142216, −6.367508736470550, −5.789081390188859, −5.443656374938854, −5.009271327616301, −4.073216728273521, −3.809088276582558, −3.298189813010617, −2.548061471810590, −1.935962635877672, −1.568810092388924, −0.8745047586017486, 0,
0.8745047586017486, 1.568810092388924, 1.935962635877672, 2.548061471810590, 3.298189813010617, 3.809088276582558, 4.073216728273521, 5.009271327616301, 5.443656374938854, 5.789081390188859, 6.367508736470550, 6.585976413142216, 7.332403378633052, 7.891072036909860, 8.314685175031064, 8.776449580161769, 9.153484649892942, 9.749882885142702, 10.22748038497542, 10.55267354615699, 10.96301871306526, 11.44342568641546, 12.01833102264666, 12.58147860193869, 12.87831275644477
