L(s) = 1 | − 3-s − 3·5-s + 4·7-s − 2·9-s − 11-s + 2·13-s + 3·15-s − 19-s − 4·21-s − 3·23-s + 4·25-s + 5·27-s − 6·29-s + 7·31-s + 33-s − 12·35-s − 7·37-s − 2·39-s + 10·43-s + 6·45-s + 9·49-s + 6·53-s + 3·55-s + 57-s − 3·59-s − 10·61-s − 8·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1.51·7-s − 2/3·9-s − 0.301·11-s + 0.554·13-s + 0.774·15-s − 0.229·19-s − 0.872·21-s − 0.625·23-s + 4/5·25-s + 0.962·27-s − 1.11·29-s + 1.25·31-s + 0.174·33-s − 2.02·35-s − 1.15·37-s − 0.320·39-s + 1.52·43-s + 0.894·45-s + 9/7·49-s + 0.824·53-s + 0.404·55-s + 0.132·57-s − 0.390·59-s − 1.28·61-s − 1.00·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.032230624\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.032230624\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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
−8.486707693362242268763893004998, −7.83265896523819007415282348063, −7.43232782285076201002591930951, −6.26538349541969487189478618895, −5.53983534604301541063552106139, −4.71817299829548860307331896732, −4.14261211235940630372604232179, −3.16366629976758759216101671656, −1.91536735204928532053488088119, −0.62134358464157050449715702997,
0.62134358464157050449715702997, 1.91536735204928532053488088119, 3.16366629976758759216101671656, 4.14261211235940630372604232179, 4.71817299829548860307331896732, 5.53983534604301541063552106139, 6.26538349541969487189478618895, 7.43232782285076201002591930951, 7.83265896523819007415282348063, 8.486707693362242268763893004998