L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 11-s − 2·13-s + 16-s + 6·17-s − 20-s − 22-s − 8·23-s + 25-s + 2·26-s + 2·29-s + 4·31-s − 32-s − 6·34-s + 10·37-s + 40-s − 2·41-s − 12·43-s + 44-s + 8·46-s − 7·49-s − 50-s − 2·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 0.554·13-s + 1/4·16-s + 1.45·17-s − 0.223·20-s − 0.213·22-s − 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.371·29-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1.64·37-s + 0.158·40-s − 0.312·41-s − 1.82·43-s + 0.150·44-s + 1.17·46-s − 49-s − 0.141·50-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 357390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.407443056\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.407443056\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−12.38585160829698, −11.92431679537715, −11.55316299485702, −11.44343312399352, −10.44548804311240, −10.21683921444662, −9.838670929264522, −9.533681315045725, −8.727247017234859, −8.436382444829561, −7.948778704347798, −7.539084584482621, −7.229568531168980, −6.503533945622010, −6.019499144472824, −5.790330193304446, −4.825342260387089, −4.641663376429662, −3.885733275989523, −3.294987989297106, −2.983302555953666, −2.158483823142835, −1.699901280345688, −0.9660421825787007, −0.4033727230199165,
0.4033727230199165, 0.9660421825787007, 1.699901280345688, 2.158483823142835, 2.983302555953666, 3.294987989297106, 3.885733275989523, 4.641663376429662, 4.825342260387089, 5.790330193304446, 6.019499144472824, 6.503533945622010, 7.229568531168980, 7.539084584482621, 7.948778704347798, 8.436382444829561, 8.727247017234859, 9.533681315045725, 9.838670929264522, 10.21683921444662, 10.44548804311240, 11.44343312399352, 11.55316299485702, 11.92431679537715, 12.38585160829698