| L(s) = 1 | − 2.37·2-s + 0.653·3-s + 3.66·4-s − 2.05·5-s − 1.55·6-s + 5.08·7-s − 3.95·8-s − 2.57·9-s + 4.90·10-s − 4.57·11-s + 2.39·12-s − 1.87·13-s − 12.1·14-s − 1.34·15-s + 2.08·16-s − 0.272·17-s + 6.12·18-s + 0.546·19-s − 7.54·20-s + 3.32·21-s + 10.8·22-s + 23-s − 2.58·24-s − 0.759·25-s + 4.45·26-s − 3.64·27-s + 18.6·28-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 0.377·3-s + 1.83·4-s − 0.920·5-s − 0.634·6-s + 1.92·7-s − 1.39·8-s − 0.857·9-s + 1.54·10-s − 1.38·11-s + 0.690·12-s − 0.518·13-s − 3.23·14-s − 0.347·15-s + 0.521·16-s − 0.0660·17-s + 1.44·18-s + 0.125·19-s − 1.68·20-s + 0.725·21-s + 2.32·22-s + 0.208·23-s − 0.527·24-s − 0.151·25-s + 0.873·26-s − 0.700·27-s + 3.52·28-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 | \( 1 + 2.37T + 2T^{2} \) |
| 3 | \( 1 - 0.653T + 3T^{2} \) |
| 5 | \( 1 + 2.05T + 5T^{2} \) |
| 7 | \( 1 - 5.08T + 7T^{2} \) |
| 11 | \( 1 + 4.57T + 11T^{2} \) |
| 13 | \( 1 + 1.87T + 13T^{2} \) |
| 17 | \( 1 + 0.272T + 17T^{2} \) |
| 19 | \( 1 - 0.546T + 19T^{2} \) |
| 31 | \( 1 + 4.25T + 31T^{2} \) |
| 37 | \( 1 - 1.52T + 37T^{2} \) |
| 41 | \( 1 + 8.02T + 41T^{2} \) |
| 43 | \( 1 + 8.67T + 43T^{2} \) |
| 47 | \( 1 - 8.58T + 47T^{2} \) |
| 53 | \( 1 - 2.34T + 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 + 7.40T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + 6.20T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 + 15.9T + 83T^{2} \) |
| 89 | \( 1 - 9.38T + 89T^{2} \) |
| 97 | \( 1 + 13.2T + 97T^{2} \) |
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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
−10.13934100466255232573122203148, −8.758292873545644903072171519753, −8.474492154769963196099495312888, −7.59409661171527106556856960046, −7.44751606326608877035103346984, −5.57146616686130961514039274932, −4.56570081433726440852590879091, −2.84303195674980008237669262715, −1.75907383561332194358115607102, 0,
1.75907383561332194358115607102, 2.84303195674980008237669262715, 4.56570081433726440852590879091, 5.57146616686130961514039274932, 7.44751606326608877035103346984, 7.59409661171527106556856960046, 8.474492154769963196099495312888, 8.758292873545644903072171519753, 10.13934100466255232573122203148
