| L(s) = 1 | − 0.283·2-s − 3.23·3-s − 1.91·4-s − 1.44·5-s + 0.919·6-s + 1.11·8-s + 7.49·9-s + 0.410·10-s + 2.25·11-s + 6.21·12-s − 2.56·13-s + 4.68·15-s + 3.52·16-s + 17-s − 2.12·18-s + 7.29·19-s + 2.77·20-s − 0.639·22-s − 5.68·23-s − 3.60·24-s − 2.90·25-s + 0.728·26-s − 14.5·27-s + 6.73·29-s − 1.33·30-s − 3.19·31-s − 3.22·32-s + ⋯ |
| L(s) = 1 | − 0.200·2-s − 1.87·3-s − 0.959·4-s − 0.646·5-s + 0.375·6-s + 0.393·8-s + 2.49·9-s + 0.129·10-s + 0.679·11-s + 1.79·12-s − 0.712·13-s + 1.20·15-s + 0.880·16-s + 0.242·17-s − 0.501·18-s + 1.67·19-s + 0.620·20-s − 0.136·22-s − 1.18·23-s − 0.735·24-s − 0.581·25-s + 0.142·26-s − 2.80·27-s + 1.25·29-s − 0.242·30-s − 0.573·31-s − 0.570·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 0.283T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 11 | \( 1 - 2.25T + 11T^{2} \) |
| 13 | \( 1 + 2.56T + 13T^{2} \) |
| 19 | \( 1 - 7.29T + 19T^{2} \) |
| 23 | \( 1 + 5.68T + 23T^{2} \) |
| 29 | \( 1 - 6.73T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 6.82T + 37T^{2} \) |
| 41 | \( 1 - 2.22T + 41T^{2} \) |
| 43 | \( 1 - 10.0T + 43T^{2} \) |
| 47 | \( 1 + 6.97T + 47T^{2} \) |
| 53 | \( 1 - 6.87T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 2.22T + 61T^{2} \) |
| 67 | \( 1 + 5.43T + 67T^{2} \) |
| 71 | \( 1 + 6.40T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 1.87T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 1.44T + 97T^{2} \) |
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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
−9.939744012270901680036856130346, −9.219248889408239886558196294895, −7.84501963866469060187992166553, −7.24633631806951770736301066796, −6.10537898062213114820304603083, −5.31299345748168684921232475262, −4.54981191403947850864300068644, −3.72492466393597089763826508682, −1.21233557122408390231545213040, 0,
1.21233557122408390231545213040, 3.72492466393597089763826508682, 4.54981191403947850864300068644, 5.31299345748168684921232475262, 6.10537898062213114820304603083, 7.24633631806951770736301066796, 7.84501963866469060187992166553, 9.219248889408239886558196294895, 9.939744012270901680036856130346
