| L(s) = 1 | + 17.7·2-s + 27·3-s + 187.·4-s + 32.5·5-s + 479.·6-s + 1.67e3·7-s + 1.05e3·8-s + 729·9-s + 577.·10-s − 3.10e3·11-s + 5.05e3·12-s − 7.08e3·13-s + 2.97e4·14-s + 877.·15-s − 5.26e3·16-s + 1.87e4·17-s + 1.29e4·18-s + 3.45e4·19-s + 6.09e3·20-s + 4.51e4·21-s − 5.51e4·22-s + 1.21e4·23-s + 2.84e4·24-s − 7.70e4·25-s − 1.25e5·26-s + 1.96e4·27-s + 3.13e5·28-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 0.577·3-s + 1.46·4-s + 0.116·5-s + 0.906·6-s + 1.84·7-s + 0.728·8-s + 0.333·9-s + 0.182·10-s − 0.703·11-s + 0.845·12-s − 0.894·13-s + 2.89·14-s + 0.0671·15-s − 0.321·16-s + 0.923·17-s + 0.523·18-s + 1.15·19-s + 0.170·20-s + 1.06·21-s − 1.10·22-s + 0.208·23-s + 0.420·24-s − 0.986·25-s − 1.40·26-s + 0.192·27-s + 2.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(6.247907978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.247907978\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 27T \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 - 17.7T + 128T^{2} \) |
| 5 | \( 1 - 32.5T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.67e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.10e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 7.08e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.87e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 3.45e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 2.50e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 8.19e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.13e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.38e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 4.30e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.33e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.03e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 4.58e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.16e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.02e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.96e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.21e4T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.33e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.28e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 4.01e5T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.54e6T + 8.07e13T^{2} \) |
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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
−13.66703992018876394888502989156, −12.26725347111782982009801498523, −11.56020176286307091225171975640, −10.13020818599643572070142712641, −8.305492760711211636325405149950, −7.28072806219391908808149884563, −5.35871731897374510465462271755, −4.72867406275401722125981384351, −3.13606973911421250556285113670, −1.78269142561771628383591720792,
1.78269142561771628383591720792, 3.13606973911421250556285113670, 4.72867406275401722125981384351, 5.35871731897374510465462271755, 7.28072806219391908808149884563, 8.305492760711211636325405149950, 10.13020818599643572070142712641, 11.56020176286307091225171975640, 12.26725347111782982009801498523, 13.66703992018876394888502989156
