| L(s) = 1 | + 2-s + 2.62·3-s + 4-s + 1.76·5-s + 2.62·6-s + 8-s + 3.89·9-s + 1.76·10-s + 2.86·11-s + 2.62·12-s + 4.62·15-s + 16-s − 0.626·17-s + 3.89·18-s − 7.25·19-s + 1.76·20-s + 2.86·22-s + 5.52·23-s + 2.62·24-s − 1.89·25-s + 2.35·27-s + 3.76·29-s + 4.62·30-s + 0.626·31-s + 32-s + 7.52·33-s − 0.626·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.51·3-s + 0.5·4-s + 0.787·5-s + 1.07·6-s + 0.353·8-s + 1.29·9-s + 0.557·10-s + 0.863·11-s + 0.758·12-s + 1.19·15-s + 0.250·16-s − 0.151·17-s + 0.918·18-s − 1.66·19-s + 0.393·20-s + 0.610·22-s + 1.15·23-s + 0.536·24-s − 0.379·25-s + 0.453·27-s + 0.698·29-s + 0.844·30-s + 0.112·31-s + 0.176·32-s + 1.30·33-s − 0.107·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.382190174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.382190174\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 2.62T + 3T^{2} \) |
| 5 | \( 1 - 1.76T + 5T^{2} \) |
| 11 | \( 1 - 2.86T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 0.626T + 17T^{2} \) |
| 19 | \( 1 + 7.25T + 19T^{2} \) |
| 23 | \( 1 - 5.52T + 23T^{2} \) |
| 29 | \( 1 - 3.76T + 29T^{2} \) |
| 31 | \( 1 - 0.626T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 43 | \( 1 - 5.87T + 43T^{2} \) |
| 47 | \( 1 + 2T + 47T^{2} \) |
| 53 | \( 1 + 1.96T + 53T^{2} \) |
| 59 | \( 1 + 2.65T + 59T^{2} \) |
| 61 | \( 1 - 0.509T + 61T^{2} \) |
| 67 | \( 1 + 3.64T + 67T^{2} \) |
| 71 | \( 1 - 2.83T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 - 5.10T + 79T^{2} \) |
| 83 | \( 1 + 0.387T + 83T^{2} \) |
| 89 | \( 1 + 8.89T + 89T^{2} \) |
| 97 | \( 1 - 7.10T + 97T^{2} \) |
| 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
−8.437320796236608392184827379936, −7.83795546872732923988039534738, −6.78743860200283861596394540524, −6.41490450373664739078251397637, −5.41722069970985333566563500053, −4.38294996475111405008141994021, −3.85974523882676149186579881171, −2.85941479972507689839230260348, −2.27593158049197524015797395717, −1.41086861196777148851246708092,
1.41086861196777148851246708092, 2.27593158049197524015797395717, 2.85941479972507689839230260348, 3.85974523882676149186579881171, 4.38294996475111405008141994021, 5.41722069970985333566563500053, 6.41490450373664739078251397637, 6.78743860200283861596394540524, 7.83795546872732923988039534738, 8.437320796236608392184827379936
