| L(s) = 1 | + 2.70·2-s + 5.29·4-s − 2.21·5-s − 2.11·7-s + 8.88·8-s − 5.99·10-s − 11-s − 2.52·13-s − 5.72·14-s + 13.4·16-s − 1.86·17-s − 1.62·19-s − 11.7·20-s − 2.70·22-s − 0.232·23-s − 0.0735·25-s − 6.81·26-s − 11.2·28-s − 8.99·29-s − 0.0529·31-s + 18.4·32-s − 5.02·34-s + 4.70·35-s − 1.12·37-s − 4.38·38-s − 19.7·40-s + 8.95·41-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 2.64·4-s − 0.992·5-s − 0.800·7-s + 3.14·8-s − 1.89·10-s − 0.301·11-s − 0.699·13-s − 1.52·14-s + 3.35·16-s − 0.451·17-s − 0.372·19-s − 2.62·20-s − 0.575·22-s − 0.0484·23-s − 0.0147·25-s − 1.33·26-s − 2.11·28-s − 1.67·29-s − 0.00951·31-s + 3.26·32-s − 0.862·34-s + 0.794·35-s − 0.184·37-s − 0.710·38-s − 3.11·40-s + 1.39·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.70T + 2T^{2} \) |
| 5 | \( 1 + 2.21T + 5T^{2} \) |
| 7 | \( 1 + 2.11T + 7T^{2} \) |
| 13 | \( 1 + 2.52T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 19 | \( 1 + 1.62T + 19T^{2} \) |
| 23 | \( 1 + 0.232T + 23T^{2} \) |
| 29 | \( 1 + 8.99T + 29T^{2} \) |
| 31 | \( 1 + 0.0529T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 41 | \( 1 - 8.95T + 41T^{2} \) |
| 43 | \( 1 + 8.83T + 43T^{2} \) |
| 47 | \( 1 - 8.15T + 47T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 - 1.89T + 59T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 + 5.42T + 71T^{2} \) |
| 73 | \( 1 + 8.03T + 73T^{2} \) |
| 79 | \( 1 + 6.67T + 79T^{2} \) |
| 83 | \( 1 + 3.06T + 83T^{2} \) |
| 89 | \( 1 + 5.40T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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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
−7.42786685746470385807831466184, −6.88632341700712852375743686141, −6.11455512667272615668129674053, −5.50403880465035192104666412090, −4.63799858730541116004489408097, −4.09896110210225630981688947933, −3.42290980106995744314526488110, −2.75531136660064718485814584541, −1.85716297397361425505249985642, 0,
1.85716297397361425505249985642, 2.75531136660064718485814584541, 3.42290980106995744314526488110, 4.09896110210225630981688947933, 4.63799858730541116004489408097, 5.50403880465035192104666412090, 6.11455512667272615668129674053, 6.88632341700712852375743686141, 7.42786685746470385807831466184
