| L(s) = 1 | + 2·3-s + 3·7-s + 9-s − 3·11-s + 4·13-s − 5·17-s − 19-s + 6·21-s − 4·27-s + 2·29-s + 8·31-s − 6·33-s + 10·37-s + 8·39-s + 6·41-s + 7·43-s + 9·47-s + 2·49-s − 10·51-s + 8·53-s − 2·57-s + 14·59-s − 5·61-s + 3·63-s − 6·71-s + 15·73-s − 9·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.13·7-s + 1/3·9-s − 0.904·11-s + 1.10·13-s − 1.21·17-s − 0.229·19-s + 1.30·21-s − 0.769·27-s + 0.371·29-s + 1.43·31-s − 1.04·33-s + 1.64·37-s + 1.28·39-s + 0.937·41-s + 1.06·43-s + 1.31·47-s + 2/7·49-s − 1.40·51-s + 1.09·53-s − 0.264·57-s + 1.82·59-s − 0.640·61-s + 0.377·63-s − 0.712·71-s + 1.75·73-s − 1.02·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.260765715\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.260765715\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
| 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
−8.414582955264067686983041740575, −8.028492410101591594332322786259, −7.30631426672271500705944975438, −6.26941543652360656448659893837, −5.49688238755594549701590373871, −4.45407740756209649247926053403, −3.97482264748533854478552219969, −2.66206605227034342400917234324, −2.33104194056647034700915811860, −1.03272483020153909426820847379,
1.03272483020153909426820847379, 2.33104194056647034700915811860, 2.66206605227034342400917234324, 3.97482264748533854478552219969, 4.45407740756209649247926053403, 5.49688238755594549701590373871, 6.26941543652360656448659893837, 7.30631426672271500705944975438, 8.028492410101591594332322786259, 8.414582955264067686983041740575
