| L(s) = 1 | − 2·5-s + 11-s + 2·17-s − 6·19-s + 6·23-s − 25-s + 2·29-s − 2·37-s − 2·41-s − 6·43-s + 6·47-s − 7·49-s + 2·53-s − 2·55-s + 8·61-s − 12·67-s − 6·71-s + 6·73-s − 12·79-s − 12·83-s − 4·85-s − 8·89-s + 12·95-s + 6·97-s − 2·101-s − 12·107-s + 4·113-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.301·11-s + 0.485·17-s − 1.37·19-s + 1.25·23-s − 1/5·25-s + 0.371·29-s − 0.328·37-s − 0.312·41-s − 0.914·43-s + 0.875·47-s − 49-s + 0.274·53-s − 0.269·55-s + 1.02·61-s − 1.46·67-s − 0.712·71-s + 0.702·73-s − 1.35·79-s − 1.31·83-s − 0.433·85-s − 0.847·89-s + 1.23·95-s + 0.609·97-s − 0.199·101-s − 1.16·107-s + 0.376·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.422830637318465495529175571444, −7.52836968068358160938962254123, −6.90231048077484383576044392193, −6.11937289864350764583968354951, −5.12891869990477172704763936246, −4.32434755255141440842877247524, −3.61400534899424541377714722299, −2.68380740919008903645509592858, −1.39742958097402412766819249463, 0,
1.39742958097402412766819249463, 2.68380740919008903645509592858, 3.61400534899424541377714722299, 4.32434755255141440842877247524, 5.12891869990477172704763936246, 6.11937289864350764583968354951, 6.90231048077484383576044392193, 7.52836968068358160938962254123, 8.422830637318465495529175571444
