L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s − 4·11-s − 13-s + 2·14-s + 16-s − 6·19-s − 20-s + 4·22-s + 4·23-s + 25-s + 26-s − 2·28-s − 6·29-s + 8·31-s − 32-s + 2·35-s − 2·37-s + 6·38-s + 40-s − 6·41-s − 2·43-s − 4·44-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 − 1.20·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.37·19-s − 0.223·20-s + 0.852·22-s + 0.834·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.338·35-s − 0.328·37-s + 0.973·38-s + 0.158·40-s − 0.937·41-s − 0.304·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.84164310848498, −12.70150912693340, −12.09398150587775, −11.53718930148017, −11.23012566670205, −10.54224321203458, −10.37633392015381, −9.911680282510073, −9.391352596478482, −8.917951896635934, −8.336538601773638, −8.120913663180380, −7.609371589489929, −6.951379218118767, −6.703942484310517, −6.242877207860130, −5.498182599922934, −5.140766668337437, −4.513789781425561, −3.944925974129107, −3.267213291310566, −2.888358364135142, −2.329656730307973, −1.757769704522896, −0.9264885105823531, 0, 0,
0.9264885105823531, 1.757769704522896, 2.329656730307973, 2.888358364135142, 3.267213291310566, 3.944925974129107, 4.513789781425561, 5.140766668337437, 5.498182599922934, 6.242877207860130, 6.703942484310517, 6.951379218118767, 7.609371589489929, 8.120913663180380, 8.336538601773638, 8.917951896635934, 9.391352596478482, 9.911680282510073, 10.37633392015381, 10.54224321203458, 11.23012566670205, 11.53718930148017, 12.09398150587775, 12.70150912693340, 12.84164310848498