| L(s) = 1 | + 2-s + 4-s + 5-s + 2·7-s + 8-s + 10-s + 3·11-s − 4·13-s + 2·14-s + 16-s + 3·17-s − 19-s + 20-s + 3·22-s + 6·23-s + 25-s − 4·26-s + 2·28-s − 6·29-s − 4·31-s + 32-s + 3·34-s + 2·35-s + 8·37-s − 38-s + 40-s + 9·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s + 0.353·8-s + 0.316·10-s + 0.904·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.229·19-s + 0.223·20-s + 0.639·22-s + 1.25·23-s + 1/5·25-s − 0.784·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s + 0.338·35-s + 1.31·37-s − 0.162·38-s + 0.158·40-s + 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 166410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.300268965\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.300268965\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.25602377157253, −12.80697248918639, −12.25324416677563, −11.96280779853486, −11.38242515317869, −10.91022495376148, −10.63323007507663, −9.819583168536234, −9.413178542040097, −9.098485171570649, −8.383917405904034, −7.768019397369977, −7.258290331918873, −7.056397726138306, −6.256730069531848, −5.712802871455056, −5.410835112709584, −4.784578135393826, −4.259487270917390, −3.883524364925361, −2.972363782298788, −2.643438037168341, −1.863568145388710, −1.386684289480718, −0.6456907424571472,
0.6456907424571472, 1.386684289480718, 1.863568145388710, 2.643438037168341, 2.972363782298788, 3.883524364925361, 4.259487270917390, 4.784578135393826, 5.410835112709584, 5.712802871455056, 6.256730069531848, 7.056397726138306, 7.258290331918873, 7.768019397369977, 8.383917405904034, 9.098485171570649, 9.413178542040097, 9.819583168536234, 10.63323007507663, 10.91022495376148, 11.38242515317869, 11.96280779853486, 12.25324416677563, 12.80697248918639, 13.25602377157253
