| L(s) = 1 | + 2.00·2-s + 1.75·3-s + 2.03·4-s − 0.905·5-s + 3.53·6-s + 0.0686·8-s + 0.0942·9-s − 1.81·10-s − 0.716·11-s + 3.57·12-s − 1.59·15-s − 3.93·16-s + 2.35·17-s + 0.189·18-s − 6.63·19-s − 1.84·20-s − 1.43·22-s + 3.75·23-s + 0.120·24-s − 4.17·25-s − 5.11·27-s + 3.25·29-s − 3.20·30-s − 1.57·31-s − 8.03·32-s − 1.26·33-s + 4.72·34-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 1.01·3-s + 1.01·4-s − 0.405·5-s + 1.44·6-s + 0.0242·8-s + 0.0314·9-s − 0.575·10-s − 0.215·11-s + 1.03·12-s − 0.411·15-s − 0.982·16-s + 0.570·17-s + 0.0446·18-s − 1.52·19-s − 0.411·20-s − 0.306·22-s + 0.783·23-s + 0.0246·24-s − 0.835·25-s − 0.983·27-s + 0.604·29-s − 0.584·30-s − 0.282·31-s − 1.41·32-s − 0.219·33-s + 0.810·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.00T + 2T^{2} \) |
| 3 | \( 1 - 1.75T + 3T^{2} \) |
| 5 | \( 1 + 0.905T + 5T^{2} \) |
| 11 | \( 1 + 0.716T + 11T^{2} \) |
| 17 | \( 1 - 2.35T + 17T^{2} \) |
| 19 | \( 1 + 6.63T + 19T^{2} \) |
| 23 | \( 1 - 3.75T + 23T^{2} \) |
| 29 | \( 1 - 3.25T + 29T^{2} \) |
| 31 | \( 1 + 1.57T + 31T^{2} \) |
| 37 | \( 1 + 5.20T + 37T^{2} \) |
| 41 | \( 1 + 4.92T + 41T^{2} \) |
| 43 | \( 1 + 9.43T + 43T^{2} \) |
| 47 | \( 1 - 8.31T + 47T^{2} \) |
| 53 | \( 1 - 14.0T + 53T^{2} \) |
| 59 | \( 1 + 0.716T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 9.39T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 3.47T + 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 3.54T + 83T^{2} \) |
| 89 | \( 1 + 12.0T + 89T^{2} \) |
| 97 | \( 1 + 7.43T + 97T^{2} \) |
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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
−7.35360270630909923231793224646, −6.70765264920119373810680074917, −5.89392104923497498807502462166, −5.28775488414796505507416397339, −4.45092302787211012556151664671, −3.86423039121182627644882553609, −3.22845578793350396747517473944, −2.61321866803953122754878859460, −1.78820638119764745327803755263, 0,
1.78820638119764745327803755263, 2.61321866803953122754878859460, 3.22845578793350396747517473944, 3.86423039121182627644882553609, 4.45092302787211012556151664671, 5.28775488414796505507416397339, 5.89392104923497498807502462166, 6.70765264920119373810680074917, 7.35360270630909923231793224646
