L(s) = 1 | + 2-s − 4-s − 5-s + 7-s − 3·8-s − 10-s + 11-s − 2·13-s + 14-s − 16-s − 2·17-s + 4·19-s + 20-s + 22-s + 25-s − 2·26-s − 28-s − 6·29-s + 5·32-s − 2·34-s − 35-s + 6·37-s + 4·38-s + 3·40-s + 6·41-s − 4·43-s − 44-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.377·7-s − 1.06·8-s − 0.316·10-s + 0.301·11-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.223·20-s + 0.213·22-s + 1/5·25-s − 0.392·26-s − 0.188·28-s − 1.11·29-s + 0.883·32-s − 0.342·34-s − 0.169·35-s + 0.986·37-s + 0.648·38-s + 0.474·40-s + 0.937·41-s − 0.609·43-s − 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.811910978\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.811910978\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.555859498553290276704555787484, −7.83555173559971230224277681079, −7.11217527582160440345781712626, −6.18029516368424477084590862750, −5.37138471604049942655746702399, −4.74255077097332149221344942072, −4.00525588330786475834031919834, −3.29030065578899668012145895978, −2.22118332277137870549461297458, −0.71217639664769246961519114732,
0.71217639664769246961519114732, 2.22118332277137870549461297458, 3.29030065578899668012145895978, 4.00525588330786475834031919834, 4.74255077097332149221344942072, 5.37138471604049942655746702399, 6.18029516368424477084590862750, 7.11217527582160440345781712626, 7.83555173559971230224277681079, 8.555859498553290276704555787484