| L(s) = 1 | + 2.17·2-s − 3.44·3-s + 2.72·4-s − 1.26·5-s − 7.49·6-s + 1.97·7-s + 1.58·8-s + 8.88·9-s − 2.74·10-s − 2.80·11-s − 9.40·12-s − 4.91·13-s + 4.29·14-s + 4.34·15-s − 2.01·16-s − 7.61·17-s + 19.3·18-s + 0.948·19-s − 3.44·20-s − 6.80·21-s − 6.10·22-s + 23-s − 5.46·24-s − 3.41·25-s − 10.6·26-s − 20.2·27-s + 5.38·28-s + ⋯ |
| L(s) = 1 | + 1.53·2-s − 1.99·3-s + 1.36·4-s − 0.563·5-s − 3.06·6-s + 0.745·7-s + 0.560·8-s + 2.96·9-s − 0.866·10-s − 0.845·11-s − 2.71·12-s − 1.36·13-s + 1.14·14-s + 1.12·15-s − 0.502·16-s − 1.84·17-s + 4.55·18-s + 0.217·19-s − 0.769·20-s − 1.48·21-s − 1.30·22-s + 0.208·23-s − 1.11·24-s − 0.682·25-s − 2.09·26-s − 3.90·27-s + 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.17T + 2T^{2} \) |
| 3 | \( 1 + 3.44T + 3T^{2} \) |
| 5 | \( 1 + 1.26T + 5T^{2} \) |
| 7 | \( 1 - 1.97T + 7T^{2} \) |
| 11 | \( 1 + 2.80T + 11T^{2} \) |
| 13 | \( 1 + 4.91T + 13T^{2} \) |
| 17 | \( 1 + 7.61T + 17T^{2} \) |
| 19 | \( 1 - 0.948T + 19T^{2} \) |
| 31 | \( 1 - 0.780T + 31T^{2} \) |
| 37 | \( 1 - 2.59T + 37T^{2} \) |
| 41 | \( 1 - 1.16T + 41T^{2} \) |
| 43 | \( 1 + 3.06T + 43T^{2} \) |
| 47 | \( 1 - 5.32T + 47T^{2} \) |
| 53 | \( 1 + 9.00T + 53T^{2} \) |
| 59 | \( 1 + 8.64T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 2.24T + 67T^{2} \) |
| 71 | \( 1 - 13.3T + 71T^{2} \) |
| 73 | \( 1 - 1.58T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 9.44T + 89T^{2} \) |
| 97 | \( 1 + 8.45T + 97T^{2} \) |
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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
−10.73321432218421900612338739715, −9.594745059555272245382043632433, −7.80567880770374491557746624452, −6.97413730669215954325032947393, −6.22337091215392514179715161984, −5.12566049769545388798994897400, −4.84917351030799214216776684036, −4.07309947305707827805939641716, −2.21912240486185377314677297559, 0,
2.21912240486185377314677297559, 4.07309947305707827805939641716, 4.84917351030799214216776684036, 5.12566049769545388798994897400, 6.22337091215392514179715161984, 6.97413730669215954325032947393, 7.80567880770374491557746624452, 9.594745059555272245382043632433, 10.73321432218421900612338739715
