| L(s) = 1 | − 0.488·2-s + 3-s − 1.76·4-s − 0.488·6-s − 5.10·7-s + 1.83·8-s + 9-s − 1.76·12-s − 1.81·13-s + 2.49·14-s + 2.62·16-s − 0.639·17-s − 0.488·18-s − 4.39·19-s − 5.10·21-s + 4.37·23-s + 1.83·24-s + 0.887·26-s + 27-s + 8.99·28-s + 7.56·29-s + 2.20·31-s − 4.95·32-s + 0.312·34-s − 1.76·36-s + 5.97·37-s + 2.14·38-s + ⋯ |
| L(s) = 1 | − 0.345·2-s + 0.577·3-s − 0.880·4-s − 0.199·6-s − 1.93·7-s + 0.649·8-s + 0.333·9-s − 0.508·12-s − 0.504·13-s + 0.666·14-s + 0.656·16-s − 0.155·17-s − 0.115·18-s − 1.00·19-s − 1.11·21-s + 0.913·23-s + 0.374·24-s + 0.174·26-s + 0.192·27-s + 1.70·28-s + 1.40·29-s + 0.396·31-s − 0.876·32-s + 0.0535·34-s − 0.293·36-s + 0.981·37-s + 0.348·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.488T + 2T^{2} \) |
| 7 | \( 1 + 5.10T + 7T^{2} \) |
| 13 | \( 1 + 1.81T + 13T^{2} \) |
| 17 | \( 1 + 0.639T + 17T^{2} \) |
| 19 | \( 1 + 4.39T + 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 - 7.56T + 29T^{2} \) |
| 31 | \( 1 - 2.20T + 31T^{2} \) |
| 37 | \( 1 - 5.97T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 + 3.38T + 43T^{2} \) |
| 47 | \( 1 + 1.51T + 47T^{2} \) |
| 53 | \( 1 + 9.17T + 53T^{2} \) |
| 59 | \( 1 - 5.37T + 59T^{2} \) |
| 61 | \( 1 - 7.26T + 61T^{2} \) |
| 67 | \( 1 + 7.25T + 67T^{2} \) |
| 71 | \( 1 - 5.50T + 71T^{2} \) |
| 73 | \( 1 - 7.14T + 73T^{2} \) |
| 79 | \( 1 - 1.90T + 79T^{2} \) |
| 83 | \( 1 + 0.161T + 83T^{2} \) |
| 89 | \( 1 + 1.19T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−7.47188694173997826014253243711, −6.62408819815322375884061785306, −6.32942126312636042043267949342, −5.20273799133407963352352190258, −4.49476474491267331638121808798, −3.75744820086080119269898438095, −3.07099941063210230261477988586, −2.38418966888173734395520163525, −0.965653644055694483071266566069, 0,
0.965653644055694483071266566069, 2.38418966888173734395520163525, 3.07099941063210230261477988586, 3.75744820086080119269898438095, 4.49476474491267331638121808798, 5.20273799133407963352352190258, 6.32942126312636042043267949342, 6.62408819815322375884061785306, 7.47188694173997826014253243711
