L(s) = 1 | + 2-s − 4-s + 5-s − 3·8-s + 10-s + 4·11-s + 13-s − 16-s − 4·19-s − 20-s + 4·22-s + 8·23-s + 25-s + 26-s − 2·29-s + 8·31-s + 5·32-s − 6·37-s − 4·38-s − 3·40-s − 6·41-s − 4·43-s − 4·44-s + 8·46-s + 8·47-s − 7·49-s + 50-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.06·8-s + 0.316·10-s + 1.20·11-s + 0.277·13-s − 1/4·16-s − 0.917·19-s − 0.223·20-s + 0.852·22-s + 1.66·23-s + 1/5·25-s + 0.196·26-s − 0.371·29-s + 1.43·31-s + 0.883·32-s − 0.986·37-s − 0.648·38-s − 0.474·40-s − 0.937·41-s − 0.609·43-s − 0.603·44-s + 1.17·46-s + 1.16·47-s − 49-s + 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169065 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.302681154\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.302681154\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−13.20311715939960, −12.88652203010459, −12.41299258274863, −11.82163722323788, −11.54693963521605, −10.88610134071346, −10.41006725788082, −9.753358831537873, −9.460669050402308, −8.849383979756580, −8.530047097538333, −8.167713466353491, −7.110456226214580, −6.758603774702337, −6.393151771173792, −5.789926186242671, −5.240847771530494, −4.828005463866192, −4.186586209000078, −3.859535313694134, −3.107811786439219, −2.769779356651398, −1.811469959011175, −1.255353619034259, −0.4931003985008987,
0.4931003985008987, 1.255353619034259, 1.811469959011175, 2.769779356651398, 3.107811786439219, 3.859535313694134, 4.186586209000078, 4.828005463866192, 5.240847771530494, 5.789926186242671, 6.393151771173792, 6.758603774702337, 7.110456226214580, 8.167713466353491, 8.530047097538333, 8.849383979756580, 9.460669050402308, 9.753358831537873, 10.41006725788082, 10.88610134071346, 11.54693963521605, 11.82163722323788, 12.41299258274863, 12.88652203010459, 13.20311715939960