| L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 11-s + 6·13-s − 15-s − 2·17-s − 4·19-s + 21-s − 8·23-s + 25-s + 27-s − 6·29-s + 33-s − 35-s − 2·37-s + 6·39-s − 10·41-s + 4·43-s − 45-s + 49-s − 2·51-s − 14·53-s − 55-s − 4·57-s − 8·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 1.66·13-s − 0.258·15-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.174·33-s − 0.169·35-s − 0.328·37-s + 0.960·39-s − 1.56·41-s + 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s − 1.92·53-s − 0.134·55-s − 0.529·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.60611609041207327104548705170, −6.56378861080354327530479823860, −6.22848394029643288058334582549, −5.28752139601497132782193373823, −4.28266486185429611548092085298, −3.89954419434393456487551643722, −3.20439722700259318145573797217, −2.03852710133780549982304565448, −1.46344647389831126755391746492, 0,
1.46344647389831126755391746492, 2.03852710133780549982304565448, 3.20439722700259318145573797217, 3.89954419434393456487551643722, 4.28266486185429611548092085298, 5.28752139601497132782193373823, 6.22848394029643288058334582549, 6.56378861080354327530479823860, 7.60611609041207327104548705170
