| L(s) = 1 | − 2-s − 0.334·3-s + 4-s + 0.808·5-s + 0.334·6-s − 8-s − 2.88·9-s − 0.808·10-s + 4.57·11-s − 0.334·12-s − 5.03·13-s − 0.270·15-s + 16-s − 17-s + 2.88·18-s + 1.53·19-s + 0.808·20-s − 4.57·22-s + 1.64·23-s + 0.334·24-s − 4.34·25-s + 5.03·26-s + 1.97·27-s − 4.02·29-s + 0.270·30-s − 0.856·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.193·3-s + 0.5·4-s + 0.361·5-s + 0.136·6-s − 0.353·8-s − 0.962·9-s − 0.255·10-s + 1.38·11-s − 0.0966·12-s − 1.39·13-s − 0.0699·15-s + 0.250·16-s − 0.242·17-s + 0.680·18-s + 0.351·19-s + 0.180·20-s − 0.975·22-s + 0.343·23-s + 0.0683·24-s − 0.869·25-s + 0.986·26-s + 0.379·27-s − 0.747·29-s + 0.0494·30-s − 0.153·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 0.334T + 3T^{2} \) |
| 5 | \( 1 - 0.808T + 5T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 + 5.03T + 13T^{2} \) |
| 19 | \( 1 - 1.53T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 + 4.02T + 29T^{2} \) |
| 31 | \( 1 + 0.856T + 31T^{2} \) |
| 37 | \( 1 + 0.0547T + 37T^{2} \) |
| 41 | \( 1 + 10.0T + 41T^{2} \) |
| 43 | \( 1 + 7.27T + 43T^{2} \) |
| 47 | \( 1 - 2.91T + 47T^{2} \) |
| 53 | \( 1 - 13.5T + 53T^{2} \) |
| 59 | \( 1 + 9.41T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 9.37T + 71T^{2} \) |
| 73 | \( 1 - 4.95T + 73T^{2} \) |
| 79 | \( 1 - 0.541T + 79T^{2} \) |
| 83 | \( 1 + 14.9T + 83T^{2} \) |
| 89 | \( 1 + 1.63T + 89T^{2} \) |
| 97 | \( 1 - 1.23T + 97T^{2} \) |
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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
−9.087543323741813840481575971863, −8.325402867623327186414434991204, −7.33607026503208823303511740069, −6.68267997342182886929357409587, −5.81697476492055622350206105895, −5.01502937697487766788148616076, −3.74766459415317498672760337462, −2.64166163099492915340439911448, −1.58816180224118257127058426747, 0,
1.58816180224118257127058426747, 2.64166163099492915340439911448, 3.74766459415317498672760337462, 5.01502937697487766788148616076, 5.81697476492055622350206105895, 6.68267997342182886929357409587, 7.33607026503208823303511740069, 8.325402867623327186414434991204, 9.087543323741813840481575971863
