| L(s) = 1 | − 3-s + 2·5-s + 4·7-s + 9-s + 11-s − 6·13-s − 2·15-s + 6·17-s + 8·19-s − 4·21-s − 25-s − 27-s + 6·29-s − 33-s + 8·35-s − 6·37-s + 6·39-s − 10·41-s + 8·43-s + 2·45-s + 9·49-s − 6·51-s − 6·53-s + 2·55-s − 8·57-s − 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.66·13-s − 0.516·15-s + 1.45·17-s + 1.83·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 0.174·33-s + 1.35·35-s − 0.986·37-s + 0.960·39-s − 1.56·41-s + 1.21·43-s + 0.298·45-s + 9/7·49-s − 0.840·51-s − 0.824·53-s + 0.269·55-s − 1.05·57-s − 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.175303541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.175303541\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.339141136925654439207262533263, −8.136326692066684045105792921301, −7.56089293819005546804727949953, −6.82430943807879406400182869542, −5.56993690843411272942663805412, −5.27293036458461851477097267556, −4.57486609696928672939637526621, −3.16578625772066727003923663964, −1.96689570329397719651168220011, −1.10197409729490824142182868653,
1.10197409729490824142182868653, 1.96689570329397719651168220011, 3.16578625772066727003923663964, 4.57486609696928672939637526621, 5.27293036458461851477097267556, 5.56993690843411272942663805412, 6.82430943807879406400182869542, 7.56089293819005546804727949953, 8.136326692066684045105792921301, 9.339141136925654439207262533263
