| L(s) = 1 | − 3-s + 4·7-s + 9-s − 2·11-s + 13-s − 6·17-s − 4·19-s − 4·21-s − 4·23-s − 5·25-s − 27-s − 10·29-s + 8·31-s + 2·33-s + 2·37-s − 39-s − 4·43-s − 2·47-s + 9·49-s + 6·51-s + 2·53-s + 4·57-s + 10·59-s − 10·61-s + 4·63-s + 8·67-s + 4·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 1.45·17-s − 0.917·19-s − 0.872·21-s − 0.834·23-s − 25-s − 0.192·27-s − 1.85·29-s + 1.43·31-s + 0.348·33-s + 0.328·37-s − 0.160·39-s − 0.609·43-s − 0.291·47-s + 9/7·49-s + 0.840·51-s + 0.274·53-s + 0.529·57-s + 1.30·59-s − 1.28·61-s + 0.503·63-s + 0.977·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−8.324902971952138704605785202150, −7.939260681586206397190079639535, −7.00034310960812354176480720233, −6.11140808730305310116200445706, −5.39682733407529528593800657224, −4.53465105360237919334130772304, −4.01593017646071294816935896556, −2.36195647663830288126441129549, −1.64247750087938000791019969761, 0,
1.64247750087938000791019969761, 2.36195647663830288126441129549, 4.01593017646071294816935896556, 4.53465105360237919334130772304, 5.39682733407529528593800657224, 6.11140808730305310116200445706, 7.00034310960812354176480720233, 7.939260681586206397190079639535, 8.324902971952138704605785202150
