| L(s) = 1 | + 1.67e5·3-s + 3.35e6·5-s − 7.07e8·7-s + 1.76e10·9-s − 8.75e10·11-s − 7.41e11·13-s + 5.62e11·15-s − 6.82e12·17-s − 5.17e13·19-s − 1.18e14·21-s + 3.13e14·23-s − 4.65e14·25-s + 1.20e15·27-s + 1.46e15·29-s + 6.42e15·31-s − 1.46e16·33-s − 2.37e15·35-s − 6.93e15·37-s − 1.24e17·39-s + 3.25e16·41-s − 4.37e16·43-s + 5.92e16·45-s + 5.34e16·47-s − 5.75e16·49-s − 1.14e18·51-s − 1.19e18·53-s − 2.93e17·55-s + ⋯ |
| L(s) = 1 | + 1.63·3-s + 0.153·5-s − 0.947·7-s + 1.68·9-s − 1.01·11-s − 1.49·13-s + 0.251·15-s − 0.821·17-s − 1.93·19-s − 1.55·21-s + 1.57·23-s − 0.976·25-s + 1.12·27-s + 0.646·29-s + 1.40·31-s − 1.66·33-s − 0.145·35-s − 0.237·37-s − 2.44·39-s + 0.378·41-s − 0.308·43-s + 0.259·45-s + 0.148·47-s − 0.103·49-s − 1.34·51-s − 0.936·53-s − 0.156·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 1.67e5T + 1.04e10T^{2} \) |
| 5 | \( 1 - 3.35e6T + 4.76e14T^{2} \) |
| 7 | \( 1 + 7.07e8T + 5.58e17T^{2} \) |
| 11 | \( 1 + 8.75e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.41e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 6.82e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 5.17e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.13e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.46e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 6.42e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 6.93e15T + 8.55e32T^{2} \) |
| 41 | \( 1 - 3.25e16T + 7.38e33T^{2} \) |
| 43 | \( 1 + 4.37e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 5.34e16T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.19e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 4.48e17T + 1.54e37T^{2} \) |
| 61 | \( 1 + 9.05e17T + 3.10e37T^{2} \) |
| 67 | \( 1 - 6.06e17T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.18e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 6.65e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 1.81e18T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.58e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.80e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 4.09e20T + 5.27e41T^{2} \) |
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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
−13.45906932523160093154463803236, −12.70720385999802883374385701659, −10.32043865117306607723089240625, −9.272178043113704104081834019054, −8.091996836514562315245382812832, −6.75671909497837027334904803762, −4.53716134348450796828322004191, −2.92542102874866213268231314547, −2.24563102678534693533129429308, 0,
2.24563102678534693533129429308, 2.92542102874866213268231314547, 4.53716134348450796828322004191, 6.75671909497837027334904803762, 8.091996836514562315245382812832, 9.272178043113704104081834019054, 10.32043865117306607723089240625, 12.70720385999802883374385701659, 13.45906932523160093154463803236
