| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 7-s + 8-s − 2·9-s − 10-s − 11-s + 12-s + 2·13-s − 14-s − 15-s + 16-s − 3·17-s − 2·18-s − 19-s − 20-s − 21-s − 22-s + 6·23-s + 24-s + 25-s + 2·26-s − 5·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.301·11-s + 0.288·12-s + 0.554·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s − 0.471·18-s − 0.229·19-s − 0.223·20-s − 0.218·21-s − 0.213·22-s + 1.25·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.540747344\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.540747344\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−13.54712946528400016991259874830, −12.94269427172040951553295571447, −11.61430649233435892606346187577, −10.81523504438992298376035875160, −9.240245216843454718321617703983, −8.197325772660852886000765577503, −6.92339690043320565435471363407, −5.57585253794403166430858921429, −3.99189905466957435614609953053, −2.72390738347756879244011461463,
2.72390738347756879244011461463, 3.99189905466957435614609953053, 5.57585253794403166430858921429, 6.92339690043320565435471363407, 8.197325772660852886000765577503, 9.240245216843454718321617703983, 10.81523504438992298376035875160, 11.61430649233435892606346187577, 12.94269427172040951553295571447, 13.54712946528400016991259874830
