L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 3·11-s − 12-s − 2·13-s + 14-s + 16-s − 4·17-s − 18-s + 21-s − 3·22-s − 3·23-s + 24-s + 2·26-s − 27-s − 28-s − 7·29-s − 8·31-s − 32-s − 3·33-s + 4·34-s + 36-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s − 0.288·12-s − 0.554·13-s + 0.267·14-s + 1/4·16-s − 0.970·17-s − 0.235·18-s + 0.218·21-s − 0.639·22-s − 0.625·23-s + 0.204·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 1.29·29-s − 1.43·31-s − 0.176·32-s − 0.522·33-s + 0.685·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5842143582\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5842143582\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 5 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 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
−12.55548228911877, −11.81809454468469, −11.53584006015114, −11.13625281795080, −10.72169000093890, −10.14027233251735, −9.813623222381067, −9.199083782547428, −9.015554908288071, −8.580207606926934, −7.745223280608874, −7.335477671399590, −7.145281861625904, −6.482334577436520, −5.960140605583032, −5.781966146385732, −5.054443945658791, −4.367025018406831, −4.035451615927384, −3.447018270515739, −2.774997933178007, −2.020778751249010, −1.792005044212331, −0.9271029637410363, −0.2669073555581042,
0.2669073555581042, 0.9271029637410363, 1.792005044212331, 2.020778751249010, 2.774997933178007, 3.447018270515739, 4.035451615927384, 4.367025018406831, 5.054443945658791, 5.781966146385732, 5.960140605583032, 6.482334577436520, 7.145281861625904, 7.335477671399590, 7.745223280608874, 8.580207606926934, 9.015554908288071, 9.199083782547428, 9.813623222381067, 10.14027233251735, 10.72169000093890, 11.13625281795080, 11.53584006015114, 11.81809454468469, 12.55548228911877