| L(s) = 1 | + 171.·3-s + 1.53e3·5-s − 3.02e3·7-s + 9.56e3·9-s − 5.19e4·11-s − 1.66e5·13-s + 2.62e5·15-s + 8.35e4·17-s + 1.03e6·19-s − 5.17e5·21-s + 6.47e5·23-s + 4.06e5·25-s − 1.73e6·27-s + 1.01e5·29-s − 1.03e6·31-s − 8.88e6·33-s − 4.65e6·35-s − 5.58e6·37-s − 2.84e7·39-s − 4.18e6·41-s + 9.60e6·43-s + 1.46e7·45-s − 3.98e7·47-s − 3.11e7·49-s + 1.42e7·51-s + 6.22e7·53-s − 7.98e7·55-s + ⋯ |
| L(s) = 1 | + 1.21·3-s + 1.09·5-s − 0.476·7-s + 0.486·9-s − 1.07·11-s − 1.61·13-s + 1.33·15-s + 0.242·17-s + 1.82·19-s − 0.581·21-s + 0.482·23-s + 0.208·25-s − 0.626·27-s + 0.0265·29-s − 0.201·31-s − 1.30·33-s − 0.523·35-s − 0.489·37-s − 1.96·39-s − 0.231·41-s + 0.428·43-s + 0.534·45-s − 1.19·47-s − 0.772·49-s + 0.295·51-s + 1.08·53-s − 1.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 8.35e4T \) |
| good | 3 | \( 1 - 171.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.53e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.02e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.19e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.66e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 1.03e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.47e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.01e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.03e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.58e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 4.18e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 9.60e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.98e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.22e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 9.60e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.86e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.73e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.04e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.95e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.72e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.96e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.93e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.75e8T + 7.60e17T^{2} \) |
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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
−9.687635301675756651001688635933, −9.144358662202144458410325322827, −7.85646577114983071177943887318, −7.21037750541947719977720895161, −5.74259540999587661770031312493, −4.91903356037742003394407462358, −3.16135344295659164699747643224, −2.67791631203522115063856988768, −1.63920764632042377562091842802, 0,
1.63920764632042377562091842802, 2.67791631203522115063856988768, 3.16135344295659164699747643224, 4.91903356037742003394407462358, 5.74259540999587661770031312493, 7.21037750541947719977720895161, 7.85646577114983071177943887318, 9.144358662202144458410325322827, 9.687635301675756651001688635933
