L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 6·11-s − 4·13-s + 14-s + 16-s − 6·17-s + 4·19-s + 20-s + 6·22-s + 25-s + 4·26-s − 28-s − 4·31-s − 32-s + 6·34-s − 35-s + 10·37-s − 4·38-s − 40-s − 6·41-s + 10·43-s − 6·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.80·11-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.223·20-s + 1.27·22-s + 1/5·25-s + 0.784·26-s − 0.188·28-s − 0.718·31-s − 0.176·32-s + 1.02·34-s − 0.169·35-s + 1.64·37-s − 0.648·38-s − 0.158·40-s − 0.937·41-s + 1.52·43-s − 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−12.97192463009193, −12.60280028408557, −12.23040005571710, −11.52534699092850, −11.06404599879197, −10.76295133901165, −10.25334935925172, −9.825209403909257, −9.359981402810815, −9.184252073004487, −8.365885797774737, −8.040391179242794, −7.517537682840302, −7.088502168287962, −6.777541812829099, −5.884017217369950, −5.734004287684605, −5.110089457506388, −4.581942454676804, −4.114557413564180, −3.027802101909805, −2.849581665203841, −2.354256358888299, −1.822309813027748, −1.010984501279962, 0, 0,
1.010984501279962, 1.822309813027748, 2.354256358888299, 2.849581665203841, 3.027802101909805, 4.114557413564180, 4.581942454676804, 5.110089457506388, 5.734004287684605, 5.884017217369950, 6.777541812829099, 7.088502168287962, 7.517537682840302, 8.040391179242794, 8.365885797774737, 9.184252073004487, 9.359981402810815, 9.825209403909257, 10.25334935925172, 10.76295133901165, 11.06404599879197, 11.52534699092850, 12.23040005571710, 12.60280028408557, 12.97192463009193