| L(s) = 1 | − 2-s − 2.39·3-s + 4-s − 5-s + 2.39·6-s + 4.25·7-s − 8-s + 2.73·9-s + 10-s − 0.795·11-s − 2.39·12-s − 0.669·13-s − 4.25·14-s + 2.39·15-s + 16-s − 4.58·17-s − 2.73·18-s − 0.597·19-s − 20-s − 10.1·21-s + 0.795·22-s + 0.869·23-s + 2.39·24-s + 25-s + 0.669·26-s + 0.641·27-s + 4.25·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.38·3-s + 0.5·4-s − 0.447·5-s + 0.977·6-s + 1.60·7-s − 0.353·8-s + 0.910·9-s + 0.316·10-s − 0.239·11-s − 0.691·12-s − 0.185·13-s − 1.13·14-s + 0.618·15-s + 0.250·16-s − 1.11·17-s − 0.643·18-s − 0.137·19-s − 0.223·20-s − 2.22·21-s + 0.169·22-s + 0.181·23-s + 0.488·24-s + 0.200·25-s + 0.131·26-s + 0.123·27-s + 0.803·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7029426991\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7029426991\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 + 2.39T + 3T^{2} \) |
| 7 | \( 1 - 4.25T + 7T^{2} \) |
| 11 | \( 1 + 0.795T + 11T^{2} \) |
| 13 | \( 1 + 0.669T + 13T^{2} \) |
| 17 | \( 1 + 4.58T + 17T^{2} \) |
| 19 | \( 1 + 0.597T + 19T^{2} \) |
| 23 | \( 1 - 0.869T + 23T^{2} \) |
| 29 | \( 1 - 0.736T + 29T^{2} \) |
| 31 | \( 1 + 1.43T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 0.764T + 41T^{2} \) |
| 43 | \( 1 - 6.58T + 43T^{2} \) |
| 47 | \( 1 + 7.33T + 47T^{2} \) |
| 53 | \( 1 - 9.71T + 53T^{2} \) |
| 59 | \( 1 + 6.87T + 59T^{2} \) |
| 61 | \( 1 + 1.21T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 - 8.66T + 73T^{2} \) |
| 79 | \( 1 - 1.66T + 79T^{2} \) |
| 83 | \( 1 - 0.693T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 8.82T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.379382010152509880784770482096, −7.71016360115950525354379160761, −7.08327080970918424997705901116, −6.24830589320907814487984346163, −5.51966445255132943820840642702, −4.72415035173688998151994931006, −4.25122809954435074576190633683, −2.66059205872837133808633475850, −1.61600260289991878836091633687, −0.59322421862943241448241418207,
0.59322421862943241448241418207, 1.61600260289991878836091633687, 2.66059205872837133808633475850, 4.25122809954435074576190633683, 4.72415035173688998151994931006, 5.51966445255132943820840642702, 6.24830589320907814487984346163, 7.08327080970918424997705901116, 7.71016360115950525354379160761, 8.379382010152509880784770482096
