| L(s) = 1 | + 2.19·2-s − 2.83·3-s + 2.83·4-s − 6.23·6-s + 1.83·8-s + 5.03·9-s − 2.56·11-s − 8.03·12-s + 0.563·13-s − 1.63·16-s − 5.19·17-s + 11.0·18-s + 0.469·19-s − 5.63·22-s − 4.03·23-s − 5.19·24-s + 1.23·26-s − 5.76·27-s + 2.86·29-s − 3.86·31-s − 7.26·32-s + 7.26·33-s − 11.4·34-s + 14.2·36-s − 2.23·37-s + 1.03·38-s − 1.59·39-s + ⋯ |
| L(s) = 1 | + 1.55·2-s − 1.63·3-s + 1.41·4-s − 2.54·6-s + 0.648·8-s + 1.67·9-s − 0.772·11-s − 2.31·12-s + 0.156·13-s − 0.408·16-s − 1.26·17-s + 2.60·18-s + 0.107·19-s − 1.20·22-s − 0.840·23-s − 1.06·24-s + 0.242·26-s − 1.10·27-s + 0.532·29-s − 0.694·31-s − 1.28·32-s + 1.26·33-s − 1.96·34-s + 2.37·36-s − 0.366·37-s + 0.167·38-s − 0.255·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 2.19T + 2T^{2} \) |
| 3 | \( 1 + 2.83T + 3T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 - 0.563T + 13T^{2} \) |
| 17 | \( 1 + 5.19T + 17T^{2} \) |
| 19 | \( 1 - 0.469T + 19T^{2} \) |
| 23 | \( 1 + 4.03T + 23T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 + 3.86T + 31T^{2} \) |
| 37 | \( 1 + 2.23T + 37T^{2} \) |
| 41 | \( 1 - 3.30T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 9.36T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 3.96T + 59T^{2} \) |
| 61 | \( 1 + 3.86T + 61T^{2} \) |
| 67 | \( 1 - 7.66T + 67T^{2} \) |
| 71 | \( 1 + 8.73T + 71T^{2} \) |
| 73 | \( 1 - 11.3T + 73T^{2} \) |
| 79 | \( 1 - 5.15T + 79T^{2} \) |
| 83 | \( 1 + 3.13T + 83T^{2} \) |
| 89 | \( 1 + 5.57T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−9.645951998286475802822546050653, −8.308187997792128864036153113881, −7.07272465192426243564263158134, −6.42727326229189834509284384568, −5.80447580142213358258879082571, −4.97422293794167416860928945998, −4.52939308379436415354626684177, −3.38683275005977024708730161933, −2.00249211382713490139547718875, 0,
2.00249211382713490139547718875, 3.38683275005977024708730161933, 4.52939308379436415354626684177, 4.97422293794167416860928945998, 5.80447580142213358258879082571, 6.42727326229189834509284384568, 7.07272465192426243564263158134, 8.308187997792128864036153113881, 9.645951998286475802822546050653
