| L(s) = 1 | − 2-s − 4-s + 5-s + 3·8-s − 10-s + 4·13-s − 16-s + 4·19-s − 20-s + 23-s + 25-s − 4·26-s − 29-s + 2·31-s − 5·32-s − 8·37-s − 4·38-s + 3·40-s + 5·41-s + 9·43-s − 46-s − 3·47-s − 50-s − 4·52-s − 4·53-s + 58-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 0.316·10-s + 1.10·13-s − 1/4·16-s + 0.917·19-s − 0.223·20-s + 0.208·23-s + 1/5·25-s − 0.784·26-s − 0.185·29-s + 0.359·31-s − 0.883·32-s − 1.31·37-s − 0.648·38-s + 0.474·40-s + 0.780·41-s + 1.37·43-s − 0.147·46-s − 0.437·47-s − 0.141·50-s − 0.554·52-s − 0.549·53-s + 0.131·58-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.835304759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.835304759\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
| 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
−12.79301134607150, −12.49811451863117, −11.72130755102581, −11.27155609712425, −10.93802883960405, −10.23485871927299, −10.10584387616785, −9.492290959694357, −9.020285603504643, −8.763336687385731, −8.197642233086147, −7.800787412055025, −7.170962813106051, −6.827836984039058, −6.116793960808407, −5.602679277810842, −5.239512411838914, −4.639740138399221, −3.975652727971558, −3.652843971916083, −2.918260998048927, −2.279791529036323, −1.491367494283315, −1.130547801617343, −0.4737181726625456,
0.4737181726625456, 1.130547801617343, 1.491367494283315, 2.279791529036323, 2.918260998048927, 3.652843971916083, 3.975652727971558, 4.639740138399221, 5.239512411838914, 5.602679277810842, 6.116793960808407, 6.827836984039058, 7.170962813106051, 7.800787412055025, 8.197642233086147, 8.763336687385731, 9.020285603504643, 9.492290959694357, 10.10584387616785, 10.23485871927299, 10.93802883960405, 11.27155609712425, 11.72130755102581, 12.49811451863117, 12.79301134607150
