L(s) = 1 | − 2-s − 4-s − 2·5-s + 2·7-s + 3·8-s + 2·10-s − 2·13-s − 2·14-s − 16-s + 2·20-s − 25-s + 2·26-s − 2·28-s + 2·29-s − 10·31-s − 5·32-s − 4·35-s − 6·37-s − 6·40-s − 4·43-s − 12·47-s − 3·49-s + 50-s + 2·52-s + 4·53-s + 6·56-s − 2·58-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.755·7-s + 1.06·8-s + 0.632·10-s − 0.554·13-s − 0.534·14-s − 1/4·16-s + 0.447·20-s − 1/5·25-s + 0.392·26-s − 0.377·28-s + 0.371·29-s − 1.79·31-s − 0.883·32-s − 0.676·35-s − 0.986·37-s − 0.948·40-s − 0.609·43-s − 1.75·47-s − 3/7·49-s + 0.141·50-s + 0.277·52-s + 0.549·53-s + 0.801·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 71 | \( 1 - T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−10.06661358161902323299520111603, −9.187201758060706258362775829227, −8.326645058194312095596989916530, −7.77193302799179830640596490677, −6.96015447976456970468915147171, −5.34081252737237418544379260743, −4.55099165635387555793378349631, −3.54497441586113907335302788719, −1.71140406603311798110159160196, 0,
1.71140406603311798110159160196, 3.54497441586113907335302788719, 4.55099165635387555793378349631, 5.34081252737237418544379260743, 6.96015447976456970468915147171, 7.77193302799179830640596490677, 8.326645058194312095596989916530, 9.187201758060706258362775829227, 10.06661358161902323299520111603