| L(s) = 1 | − 0.193·2-s − 1.07·3-s − 1.96·4-s − 5-s + 0.207·6-s + 1.51·7-s + 0.765·8-s − 1.84·9-s + 0.193·10-s − 11-s + 2.11·12-s − 1.00·13-s − 0.292·14-s + 1.07·15-s + 3.77·16-s − 3.23·17-s + 0.356·18-s + 0.322·19-s + 1.96·20-s − 1.62·21-s + 0.193·22-s + 6.42·23-s − 0.823·24-s + 25-s + 0.194·26-s + 5.20·27-s − 2.97·28-s + ⋯ |
| L(s) = 1 | − 0.136·2-s − 0.620·3-s − 0.981·4-s − 0.447·5-s + 0.0848·6-s + 0.572·7-s + 0.270·8-s − 0.614·9-s + 0.0611·10-s − 0.301·11-s + 0.609·12-s − 0.279·13-s − 0.0781·14-s + 0.277·15-s + 0.944·16-s − 0.783·17-s + 0.0839·18-s + 0.0738·19-s + 0.438·20-s − 0.355·21-s + 0.0411·22-s + 1.33·23-s − 0.168·24-s + 0.200·25-s + 0.0382·26-s + 1.00·27-s − 0.561·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 + 0.193T + 2T^{2} \) |
| 3 | \( 1 + 1.07T + 3T^{2} \) |
| 7 | \( 1 - 1.51T + 7T^{2} \) |
| 13 | \( 1 + 1.00T + 13T^{2} \) |
| 17 | \( 1 + 3.23T + 17T^{2} \) |
| 19 | \( 1 - 0.322T + 19T^{2} \) |
| 23 | \( 1 - 6.42T + 23T^{2} \) |
| 29 | \( 1 + 0.425T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 + 1.97T + 37T^{2} \) |
| 41 | \( 1 - 9.96T + 41T^{2} \) |
| 43 | \( 1 - 2.86T + 43T^{2} \) |
| 47 | \( 1 + 5.59T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 5.30T + 59T^{2} \) |
| 61 | \( 1 - 1.49T + 61T^{2} \) |
| 67 | \( 1 - 1.04T + 67T^{2} \) |
| 71 | \( 1 + 8.30T + 71T^{2} \) |
| 79 | \( 1 - 9.14T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 - 7.78T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.096470868313790858124852129404, −7.50838773025152659964155072656, −6.60121575418036393482323751735, −5.63917860665644475188199623117, −5.02854694600311550329778177806, −4.50273834715465518022853664393, −3.54580046642915629394289916669, −2.50222293691202078247034695843, −1.03647197112371223592405966693, 0,
1.03647197112371223592405966693, 2.50222293691202078247034695843, 3.54580046642915629394289916669, 4.50273834715465518022853664393, 5.02854694600311550329778177806, 5.63917860665644475188199623117, 6.60121575418036393482323751735, 7.50838773025152659964155072656, 8.096470868313790858124852129404
