| L(s) = 1 | − 2-s + 4-s + 0.652·5-s + 0.532·7-s − 8-s − 0.652·10-s + 2.22·11-s − 6.41·13-s − 0.532·14-s + 16-s + 3.04·17-s + 0.879·19-s + 0.652·20-s − 2.22·22-s + 2.83·23-s − 4.57·25-s + 6.41·26-s + 0.532·28-s − 8.04·29-s − 7.24·31-s − 32-s − 3.04·34-s + 0.347·35-s + 10.1·37-s − 0.879·38-s − 0.652·40-s + 6.24·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.291·5-s + 0.201·7-s − 0.353·8-s − 0.206·10-s + 0.671·11-s − 1.77·13-s − 0.142·14-s + 0.250·16-s + 0.737·17-s + 0.201·19-s + 0.145·20-s − 0.474·22-s + 0.591·23-s − 0.914·25-s + 1.25·26-s + 0.100·28-s − 1.49·29-s − 1.30·31-s − 0.176·32-s − 0.521·34-s + 0.0587·35-s + 1.66·37-s − 0.142·38-s − 0.103·40-s + 0.975·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 5 | \( 1 - 0.652T + 5T^{2} \) |
| 7 | \( 1 - 0.532T + 7T^{2} \) |
| 11 | \( 1 - 2.22T + 11T^{2} \) |
| 13 | \( 1 + 6.41T + 13T^{2} \) |
| 17 | \( 1 - 3.04T + 17T^{2} \) |
| 19 | \( 1 - 0.879T + 19T^{2} \) |
| 23 | \( 1 - 2.83T + 23T^{2} \) |
| 29 | \( 1 + 8.04T + 29T^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 - 10.1T + 37T^{2} \) |
| 41 | \( 1 - 6.24T + 41T^{2} \) |
| 43 | \( 1 + 5.63T + 43T^{2} \) |
| 47 | \( 1 + 5.92T + 47T^{2} \) |
| 53 | \( 1 - 6.96T + 53T^{2} \) |
| 59 | \( 1 - 0.716T + 59T^{2} \) |
| 61 | \( 1 + 5.83T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 3T + 71T^{2} \) |
| 73 | \( 1 + 1.28T + 73T^{2} \) |
| 79 | \( 1 - 5.29T + 79T^{2} \) |
| 83 | \( 1 - 2.96T + 83T^{2} \) |
| 89 | \( 1 + 0.448T + 89T^{2} \) |
| 97 | \( 1 + 3.36T + 97T^{2} \) |
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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
−7.82932643748712263761104073303, −7.59094137148807213867702744360, −6.80876565515720870745823684546, −5.87232654099195175732320365904, −5.23869489967962269298029900678, −4.26565188646406035700807437304, −3.23457573479549653298435554648, −2.27014236200276274535337146358, −1.42088029083976353758203242934, 0,
1.42088029083976353758203242934, 2.27014236200276274535337146358, 3.23457573479549653298435554648, 4.26565188646406035700807437304, 5.23869489967962269298029900678, 5.87232654099195175732320365904, 6.80876565515720870745823684546, 7.59094137148807213867702744360, 7.82932643748712263761104073303
