| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s − 13-s − 14-s + 16-s + 17-s + 7·19-s − 20-s + 3·23-s + 25-s − 26-s − 28-s − 8·29-s − 5·31-s + 32-s + 34-s + 35-s + 5·37-s + 7·38-s − 40-s − 10·41-s − 2·43-s + 3·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.277·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 1.60·19-s − 0.223·20-s + 0.625·23-s + 1/5·25-s − 0.196·26-s − 0.188·28-s − 1.48·29-s − 0.898·31-s + 0.176·32-s + 0.171·34-s + 0.169·35-s + 0.821·37-s + 1.13·38-s − 0.158·40-s − 1.56·41-s − 0.304·43-s + 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.592518418\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.592518418\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 3 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
| 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
−13.06266995736105, −12.83496546017773, −12.12980734293734, −11.81636026080478, −11.34361301784078, −10.96858110718681, −10.40372938972236, −9.643700750264284, −9.559821094664655, −8.923564099769588, −8.196104431439932, −7.695433736776181, −7.388548062540633, −6.780484763835637, −6.419382309195276, −5.608190487497111, −5.269921756383044, −4.896827961381958, −4.111660105219992, −3.586545448599282, −3.222381761168937, −2.707801925013193, −1.848849192772758, −1.309404121239222, −0.4045827538685826,
0.4045827538685826, 1.309404121239222, 1.848849192772758, 2.707801925013193, 3.222381761168937, 3.586545448599282, 4.111660105219992, 4.896827961381958, 5.269921756383044, 5.608190487497111, 6.419382309195276, 6.780484763835637, 7.388548062540633, 7.695433736776181, 8.196104431439932, 8.923564099769588, 9.559821094664655, 9.643700750264284, 10.40372938972236, 10.96858110718681, 11.34361301784078, 11.81636026080478, 12.12980734293734, 12.83496546017773, 13.06266995736105
