L(s) = 1 | − 3-s + 9-s − 4·11-s + 5·13-s − 6·17-s + 5·19-s + 2·23-s − 27-s − 2·29-s + 9·31-s + 4·33-s + 11·37-s − 5·39-s − 8·41-s + 43-s + 6·47-s + 6·51-s + 4·53-s − 5·57-s − 6·59-s − 14·61-s − 67-s − 2·69-s − 12·71-s − 11·73-s − 79-s + 81-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.20·11-s + 1.38·13-s − 1.45·17-s + 1.14·19-s + 0.417·23-s − 0.192·27-s − 0.371·29-s + 1.61·31-s + 0.696·33-s + 1.80·37-s − 0.800·39-s − 1.24·41-s + 0.152·43-s + 0.875·47-s + 0.840·51-s + 0.549·53-s − 0.662·57-s − 0.781·59-s − 1.79·61-s − 0.122·67-s − 0.240·69-s − 1.42·71-s − 1.28·73-s − 0.112·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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
−14.71831182046020, −13.75176493320490, −13.43372155589713, −13.30974708458223, −12.63118856633648, −11.86986049844236, −11.56249022903290, −11.03736492390523, −10.48585678894773, −10.25834605189098, −9.361500120778035, −8.987489438241372, −8.349394475335405, −7.783098473237846, −7.315543891523821, −6.598168534397605, −6.062780524340585, −5.720479598957680, −4.866545664596368, −4.564038851925544, −3.832547184839333, −2.991043812855940, −2.570781318507554, −1.566810004782938, −0.9088094027824988, 0,
0.9088094027824988, 1.566810004782938, 2.570781318507554, 2.991043812855940, 3.832547184839333, 4.564038851925544, 4.866545664596368, 5.720479598957680, 6.062780524340585, 6.598168534397605, 7.315543891523821, 7.783098473237846, 8.349394475335405, 8.987489438241372, 9.361500120778035, 10.25834605189098, 10.48585678894773, 11.03736492390523, 11.56249022903290, 11.86986049844236, 12.63118856633648, 13.30974708458223, 13.43372155589713, 13.75176493320490, 14.71831182046020