L(s) = 1 | + 2-s + 4-s − 7-s + 8-s − 11-s − 4·13-s − 14-s + 16-s − 2·17-s − 22-s − 6·23-s − 4·26-s − 28-s + 2·31-s + 32-s − 2·34-s − 8·37-s − 2·41-s − 4·43-s − 44-s − 6·46-s + 8·47-s + 49-s − 4·52-s − 6·53-s − 56-s + 2·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.377·7-s + 0.353·8-s − 0.301·11-s − 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.213·22-s − 1.25·23-s − 0.784·26-s − 0.188·28-s + 0.359·31-s + 0.176·32-s − 0.342·34-s − 1.31·37-s − 0.312·41-s − 0.609·43-s − 0.150·44-s − 0.884·46-s + 1.16·47-s + 1/7·49-s − 0.554·52-s − 0.824·53-s − 0.133·56-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.941607177\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.941607177\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.03749015236393, −14.26717668267976, −13.95783141077542, −13.45909244766394, −12.79217465347756, −12.40229287882263, −11.90435739792927, −11.46664078576307, −10.64092518142379, −10.25135401905832, −9.768564792804468, −9.082017731517501, −8.441676193170019, −7.771588341511620, −7.275008303938914, −6.672272101139265, −6.140186002930359, −5.471395888877883, −4.911333447498825, −4.345138266232317, −3.655079589012884, −2.999235135875291, −2.294278220468190, −1.739647989622449, −0.4330482996901106,
0.4330482996901106, 1.739647989622449, 2.294278220468190, 2.999235135875291, 3.655079589012884, 4.345138266232317, 4.911333447498825, 5.471395888877883, 6.140186002930359, 6.672272101139265, 7.275008303938914, 7.771588341511620, 8.441676193170019, 9.082017731517501, 9.768564792804468, 10.25135401905832, 10.64092518142379, 11.46664078576307, 11.90435739792927, 12.40229287882263, 12.79217465347756, 13.45909244766394, 13.95783141077542, 14.26717668267976, 15.03749015236393