| L(s) = 1 | − 2.81·2-s − 2.68·3-s + 5.90·4-s + 2.99·5-s + 7.53·6-s + 4.86·7-s − 10.9·8-s + 4.19·9-s − 8.42·10-s − 1.33·11-s − 15.8·12-s − 3.27·13-s − 13.6·14-s − 8.03·15-s + 19.0·16-s − 4.03·17-s − 11.7·18-s + 4.45·19-s + 17.6·20-s − 13.0·21-s + 3.74·22-s − 1.35·23-s + 29.4·24-s + 3.97·25-s + 9.21·26-s − 3.20·27-s + 28.6·28-s + ⋯ |
| L(s) = 1 | − 1.98·2-s − 1.54·3-s + 2.95·4-s + 1.33·5-s + 3.07·6-s + 1.83·7-s − 3.87·8-s + 1.39·9-s − 2.66·10-s − 0.401·11-s − 4.56·12-s − 0.909·13-s − 3.65·14-s − 2.07·15-s + 4.75·16-s − 0.978·17-s − 2.77·18-s + 1.02·19-s + 3.95·20-s − 2.84·21-s + 0.797·22-s − 0.282·23-s + 6.00·24-s + 0.795·25-s + 1.80·26-s − 0.616·27-s + 5.42·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 + 2.81T + 2T^{2} \) |
| 3 | \( 1 + 2.68T + 3T^{2} \) |
| 5 | \( 1 - 2.99T + 5T^{2} \) |
| 7 | \( 1 - 4.86T + 7T^{2} \) |
| 11 | \( 1 + 1.33T + 11T^{2} \) |
| 13 | \( 1 + 3.27T + 13T^{2} \) |
| 17 | \( 1 + 4.03T + 17T^{2} \) |
| 19 | \( 1 - 4.45T + 19T^{2} \) |
| 23 | \( 1 + 1.35T + 23T^{2} \) |
| 29 | \( 1 - 5.93T + 29T^{2} \) |
| 31 | \( 1 + 2.35T + 31T^{2} \) |
| 41 | \( 1 + 7.42T + 41T^{2} \) |
| 43 | \( 1 + 4.67T + 43T^{2} \) |
| 47 | \( 1 + 7.56T + 47T^{2} \) |
| 53 | \( 1 - 2.65T + 53T^{2} \) |
| 59 | \( 1 + 5.14T + 59T^{2} \) |
| 61 | \( 1 + 13.5T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 7.04T + 73T^{2} \) |
| 79 | \( 1 - 14.1T + 79T^{2} \) |
| 83 | \( 1 - 2.53T + 83T^{2} \) |
| 89 | \( 1 - 1.25T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.079456067620148367155810534573, −7.45130352931740935153911481566, −6.70311529657367309229791558590, −6.08699628064400561528569749738, −5.27810166828616046252532268591, −4.86553983656860528252448956620, −2.69854444564170844128532611567, −1.77934478481261420645328885104, −1.32228133588697941365032451774, 0,
1.32228133588697941365032451774, 1.77934478481261420645328885104, 2.69854444564170844128532611567, 4.86553983656860528252448956620, 5.27810166828616046252532268591, 6.08699628064400561528569749738, 6.70311529657367309229791558590, 7.45130352931740935153911481566, 8.079456067620148367155810534573
