L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 5·7-s − 8-s − 2·9-s + 10-s + 11-s + 12-s + 2·13-s − 5·14-s − 15-s + 16-s + 3·17-s + 2·18-s − 7·19-s − 20-s + 5·21-s − 22-s − 6·23-s − 24-s + 25-s − 2·26-s − 5·27-s + 5·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 1.88·7-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 − 1.33·14-s − 0.258·15-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.60·19-s − 0.223·20-s + 1.09·21-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 + 0.944·28-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
\(L(1)\) |
\(\approx\) |
\(0.9420579665\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9420579665\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−14.13397147037926336250163189812, −12.37150177127699508441440919910, −11.32146429554650552999732954698, −10.70422776242305107048865394084, −8.996361573480265218453634288731, −8.285662434899332908926558240119, −7.56981196469027501153188760806, −5.73876439085829035805370640147, −4.01173135747517138761270457828, −1.97170842587956421114470880781,
1.97170842587956421114470880781, 4.01173135747517138761270457828, 5.73876439085829035805370640147, 7.56981196469027501153188760806, 8.285662434899332908926558240119, 8.996361573480265218453634288731, 10.70422776242305107048865394084, 11.32146429554650552999732954698, 12.37150177127699508441440919910, 14.13397147037926336250163189812