| L(s) = 1 | − 38.3·2-s − 247.·3-s + 958.·4-s + 875.·5-s + 9.47e3·6-s + 7.07e3·7-s − 1.71e4·8-s + 4.13e4·9-s − 3.35e4·10-s − 5.52e4·11-s − 2.36e5·12-s − 1.79e5·13-s − 2.71e5·14-s − 2.16e5·15-s + 1.65e5·16-s − 1.58e6·18-s − 5.65e4·19-s + 8.39e5·20-s − 1.74e6·21-s + 2.12e6·22-s − 2.36e6·23-s + 4.22e6·24-s − 1.18e6·25-s + 6.89e6·26-s − 5.35e6·27-s + 6.77e6·28-s + 1.38e6·29-s + ⋯ |
| L(s) = 1 | − 1.69·2-s − 1.76·3-s + 1.87·4-s + 0.626·5-s + 2.98·6-s + 1.11·7-s − 1.47·8-s + 2.10·9-s − 1.06·10-s − 1.13·11-s − 3.29·12-s − 1.74·13-s − 1.88·14-s − 1.10·15-s + 0.631·16-s − 3.55·18-s − 0.0994·19-s + 1.17·20-s − 1.96·21-s + 1.92·22-s − 1.76·23-s + 2.60·24-s − 0.607·25-s + 2.95·26-s − 1.93·27-s + 2.08·28-s + 0.364·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.01274995090\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01274995090\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + 38.3T + 512T^{2} \) |
| 3 | \( 1 + 247.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 875.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 7.07e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.52e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.79e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 5.65e4T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.36e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 1.38e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.30e5T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.73e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.41e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.24e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 4.13e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.71e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 3.33e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 5.58e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.25e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 6.66e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 2.13e6T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.27e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.63e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.86e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 9.45e7T + 7.60e17T^{2} \) |
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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
−10.20457184009915172168675124018, −9.735203773090779818407910957408, −8.186530210265034138126715530331, −7.49768668163660020916732150035, −6.53840405188261921056040894321, −5.42175719793519857651416225667, −4.76755179150761349662194737355, −2.17371565551078266229468851270, −1.54597059180218407267627591946, −0.07147558959616778084812609023,
0.07147558959616778084812609023, 1.54597059180218407267627591946, 2.17371565551078266229468851270, 4.76755179150761349662194737355, 5.42175719793519857651416225667, 6.53840405188261921056040894321, 7.49768668163660020916732150035, 8.186530210265034138126715530331, 9.735203773090779818407910957408, 10.20457184009915172168675124018
