| L(s) = 1 | + 1.55·3-s + 3.12·5-s − 0.577·9-s − 5.30·11-s + 3.11·13-s + 4.85·15-s − 17-s − 5.65·19-s − 6.92·23-s + 4.75·25-s − 5.56·27-s + 3.34·29-s − 9.32·31-s − 8.24·33-s + 0.437·37-s + 4.85·39-s − 4.48·41-s + 2.07·43-s − 1.80·45-s − 10.4·47-s − 1.55·51-s + 4.15·53-s − 16.5·55-s − 8.79·57-s + 3.68·59-s + 2.38·61-s + 9.73·65-s + ⋯ |
| L(s) = 1 | + 0.898·3-s + 1.39·5-s − 0.192·9-s − 1.59·11-s + 0.865·13-s + 1.25·15-s − 0.242·17-s − 1.29·19-s − 1.44·23-s + 0.950·25-s − 1.07·27-s + 0.621·29-s − 1.67·31-s − 1.43·33-s + 0.0719·37-s + 0.777·39-s − 0.701·41-s + 0.316·43-s − 0.268·45-s − 1.51·47-s − 0.217·51-s + 0.570·53-s − 2.23·55-s − 1.16·57-s + 0.479·59-s + 0.305·61-s + 1.20·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.55T + 3T^{2} \) |
| 5 | \( 1 - 3.12T + 5T^{2} \) |
| 11 | \( 1 + 5.30T + 11T^{2} \) |
| 13 | \( 1 - 3.11T + 13T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 - 3.34T + 29T^{2} \) |
| 31 | \( 1 + 9.32T + 31T^{2} \) |
| 37 | \( 1 - 0.437T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 - 2.07T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 4.15T + 53T^{2} \) |
| 59 | \( 1 - 3.68T + 59T^{2} \) |
| 61 | \( 1 - 2.38T + 61T^{2} \) |
| 67 | \( 1 + 1.99T + 67T^{2} \) |
| 71 | \( 1 + 11.0T + 71T^{2} \) |
| 73 | \( 1 - 0.306T + 73T^{2} \) |
| 79 | \( 1 + 4.68T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 + 11.0T + 89T^{2} \) |
| 97 | \( 1 - 4.44T + 97T^{2} \) |
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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
−7.87460326710415159935073325715, −6.93687237967293060575145283356, −5.98901952989328449786643138464, −5.74956127462946827574240935850, −4.84737817923224815292360714604, −3.84133073023448731353651896776, −2.98601344202859453158311659618, −2.19449993579561168063591517067, −1.81217062564505939973711898687, 0,
1.81217062564505939973711898687, 2.19449993579561168063591517067, 2.98601344202859453158311659618, 3.84133073023448731353651896776, 4.84737817923224815292360714604, 5.74956127462946827574240935850, 5.98901952989328449786643138464, 6.93687237967293060575145283356, 7.87460326710415159935073325715
