| L(s) = 1 | + 1.56·2-s − 3-s + 0.438·4-s − 1.56·5-s − 1.56·6-s − 2.43·8-s + 9-s − 2.43·10-s − 2·11-s − 0.438·12-s + 6.12·13-s + 1.56·15-s − 4.68·16-s + 7.56·17-s + 1.56·18-s − 1.43·19-s − 0.684·20-s − 3.12·22-s + 23-s + 2.43·24-s − 2.56·25-s + 9.56·26-s − 27-s − 9.12·29-s + 2.43·30-s − 5.68·31-s − 2.43·32-s + ⋯ |
| L(s) = 1 | + 1.10·2-s − 0.577·3-s + 0.219·4-s − 0.698·5-s − 0.637·6-s − 0.862·8-s + 0.333·9-s − 0.771·10-s − 0.603·11-s − 0.126·12-s + 1.69·13-s + 0.403·15-s − 1.17·16-s + 1.83·17-s + 0.368·18-s − 0.330·19-s − 0.153·20-s − 0.665·22-s + 0.208·23-s + 0.497·24-s − 0.512·25-s + 1.87·26-s − 0.192·27-s − 1.69·29-s + 0.445·30-s − 1.02·31-s − 0.431·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 5 | \( 1 + 1.56T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 6.12T + 13T^{2} \) |
| 17 | \( 1 - 7.56T + 17T^{2} \) |
| 19 | \( 1 + 1.43T + 19T^{2} \) |
| 29 | \( 1 + 9.12T + 29T^{2} \) |
| 31 | \( 1 + 5.68T + 31T^{2} \) |
| 37 | \( 1 - 3.43T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 + 0.315T + 43T^{2} \) |
| 47 | \( 1 + 6.68T + 47T^{2} \) |
| 53 | \( 1 + 7.80T + 53T^{2} \) |
| 59 | \( 1 + 9.12T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 5.87T + 73T^{2} \) |
| 79 | \( 1 + 5.43T + 79T^{2} \) |
| 83 | \( 1 + 4.87T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 15.3T + 97T^{2} \) |
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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
−7.946733349766086775637059846075, −7.58052148896907937317999462687, −6.35314298476445282908764172026, −5.78201037381197237855648198410, −5.30823723209052916894712358424, −4.25362927966248744727776937245, −3.70997219923970979464632497441, −3.03476599192943406912289431201, −1.44187655184521798222135422784, 0,
1.44187655184521798222135422784, 3.03476599192943406912289431201, 3.70997219923970979464632497441, 4.25362927966248744727776937245, 5.30823723209052916894712358424, 5.78201037381197237855648198410, 6.35314298476445282908764172026, 7.58052148896907937317999462687, 7.946733349766086775637059846075
