L(s) = 1 | − 1.81·2-s − 3-s + 1.28·4-s + 5-s + 1.81·6-s − 7-s + 1.28·8-s + 9-s − 1.81·10-s − 11-s − 1.28·12-s + 5.10·13-s + 1.81·14-s − 15-s − 4.91·16-s + 4.28·17-s − 1.81·18-s − 3.33·19-s + 1.28·20-s + 21-s + 1.81·22-s − 3.91·23-s − 1.28·24-s + 25-s − 9.25·26-s − 27-s − 1.28·28-s + ⋯ |
L(s) = 1 | − 1.28·2-s − 0.577·3-s + 0.644·4-s + 0.447·5-s + 0.740·6-s − 0.377·7-s + 0.455·8-s + 0.333·9-s − 0.573·10-s − 0.301·11-s − 0.372·12-s + 1.41·13-s + 0.484·14-s − 0.258·15-s − 1.22·16-s + 1.04·17-s − 0.427·18-s − 0.765·19-s + 0.288·20-s + 0.218·21-s + 0.386·22-s − 0.816·23-s − 0.263·24-s + 0.200·25-s − 1.81·26-s − 0.192·27-s − 0.243·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6782320571\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6782320571\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 + 1.81T + 2T^{2} \) |
| 13 | \( 1 - 5.10T + 13T^{2} \) |
| 17 | \( 1 - 4.28T + 17T^{2} \) |
| 19 | \( 1 + 3.33T + 19T^{2} \) |
| 23 | \( 1 + 3.91T + 23T^{2} \) |
| 29 | \( 1 - 9.01T + 29T^{2} \) |
| 31 | \( 1 + 3.68T + 31T^{2} \) |
| 37 | \( 1 + 1.10T + 37T^{2} \) |
| 41 | \( 1 + 6.72T + 41T^{2} \) |
| 43 | \( 1 + 5.39T + 43T^{2} \) |
| 47 | \( 1 - 8.72T + 47T^{2} \) |
| 53 | \( 1 - 8.75T + 53T^{2} \) |
| 59 | \( 1 + 5.39T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 3.62T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 4.96T + 83T^{2} \) |
| 89 | \( 1 - 15.1T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−10.02021698623246540151113350595, −8.894149085106993743107113148390, −8.400472573052066830496049984055, −7.46720777188228196011578225205, −6.50024684803329558637712668580, −5.85653637764560989405804031092, −4.71509751443552185785746495275, −3.50634987380567245822901997657, −1.94068935958519665405785224417, −0.796909974331526722441096830850,
0.796909974331526722441096830850, 1.94068935958519665405785224417, 3.50634987380567245822901997657, 4.71509751443552185785746495275, 5.85653637764560989405804031092, 6.50024684803329558637712668580, 7.46720777188228196011578225205, 8.400472573052066830496049984055, 8.894149085106993743107113148390, 10.02021698623246540151113350595