| L(s) = 1 | − 3·3-s + 3·5-s + 2·7-s + 6·9-s − 11-s − 3·13-s − 9·15-s − 4·17-s − 8·19-s − 6·21-s + 4·25-s − 9·27-s + 29-s − 3·31-s + 3·33-s + 6·35-s + 8·37-s + 9·39-s − 2·41-s + 7·43-s + 18·45-s − 11·47-s − 3·49-s + 12·51-s − 53-s − 3·55-s + 24·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s + 0.755·7-s + 2·9-s − 0.301·11-s − 0.832·13-s − 2.32·15-s − 0.970·17-s − 1.83·19-s − 1.30·21-s + 4/5·25-s − 1.73·27-s + 0.185·29-s − 0.538·31-s + 0.522·33-s + 1.01·35-s + 1.31·37-s + 1.44·39-s − 0.312·41-s + 1.06·43-s + 2.68·45-s − 1.60·47-s − 3/7·49-s + 1.68·51-s − 0.137·53-s − 0.404·55-s + 3.17·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.050758124201678791033621329821, −7.962357345640797971577528915374, −6.90898810262118971316359808012, −6.26668821754321326684906931279, −5.72305014285753421266225529472, −4.82000319309045720683927918981, −4.43557931381636842001263386788, −2.38846470504631823404636041964, −1.57888278458505740991793186002, 0,
1.57888278458505740991793186002, 2.38846470504631823404636041964, 4.43557931381636842001263386788, 4.82000319309045720683927918981, 5.72305014285753421266225529472, 6.26668821754321326684906931279, 6.90898810262118971316359808012, 7.962357345640797971577528915374, 9.050758124201678791033621329821
