| L(s) = 1 | + 3-s − 2·9-s + 3·11-s + 6·13-s + 5·17-s + 19-s + 7·23-s − 5·27-s + 2·29-s − 5·31-s + 3·33-s − 3·37-s + 6·39-s − 2·41-s + 4·43-s − 5·47-s + 5·51-s + 53-s + 57-s + 15·59-s − 5·61-s + 9·67-s + 7·69-s − 7·73-s + 79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s + 0.904·11-s + 1.66·13-s + 1.21·17-s + 0.229·19-s + 1.45·23-s − 0.962·27-s + 0.371·29-s − 0.898·31-s + 0.522·33-s − 0.493·37-s + 0.960·39-s − 0.312·41-s + 0.609·43-s − 0.729·47-s + 0.700·51-s + 0.137·53-s + 0.132·57-s + 1.95·59-s − 0.640·61-s + 1.09·67-s + 0.842·69-s − 0.819·73-s + 0.112·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.185161572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.185161572\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−7.72341070324985048487652292167, −7.03215146573223775757656558195, −6.28810471826707613029643426015, −5.67214494827715996314297794919, −5.01453100508223327769185214179, −3.81626556713052368107194322550, −3.51718424436917187767630716372, −2.77542030900405681818067140794, −1.61992065202571272735219686431, −0.891614962987642210491003464113,
0.891614962987642210491003464113, 1.61992065202571272735219686431, 2.77542030900405681818067140794, 3.51718424436917187767630716372, 3.81626556713052368107194322550, 5.01453100508223327769185214179, 5.67214494827715996314297794919, 6.28810471826707613029643426015, 7.03215146573223775757656558195, 7.72341070324985048487652292167
