| L(s) = 1 | − 0.647·2-s + 3-s − 1.58·4-s + 4.33·5-s − 0.647·6-s + 2.31·8-s + 9-s − 2.80·10-s + 1.61·11-s − 1.58·12-s + 2.02·13-s + 4.33·15-s + 1.66·16-s + 2.10·17-s − 0.647·18-s + 5.54·19-s − 6.85·20-s − 1.04·22-s − 5.14·23-s + 2.31·24-s + 13.8·25-s − 1.30·26-s + 27-s + 3.31·29-s − 2.80·30-s + 3.67·31-s − 5.71·32-s + ⋯ |
| L(s) = 1 | − 0.457·2-s + 0.577·3-s − 0.790·4-s + 1.93·5-s − 0.264·6-s + 0.819·8-s + 0.333·9-s − 0.887·10-s + 0.488·11-s − 0.456·12-s + 0.560·13-s + 1.11·15-s + 0.415·16-s + 0.509·17-s − 0.152·18-s + 1.27·19-s − 1.53·20-s − 0.223·22-s − 1.07·23-s + 0.473·24-s + 2.76·25-s − 0.256·26-s + 0.192·27-s + 0.615·29-s − 0.512·30-s + 0.660·31-s − 1.00·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8967 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.140054739\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.140054739\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 0.647T + 2T^{2} \) |
| 5 | \( 1 - 4.33T + 5T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 - 2.02T + 13T^{2} \) |
| 17 | \( 1 - 2.10T + 17T^{2} \) |
| 19 | \( 1 - 5.54T + 19T^{2} \) |
| 23 | \( 1 + 5.14T + 23T^{2} \) |
| 29 | \( 1 - 3.31T + 29T^{2} \) |
| 31 | \( 1 - 3.67T + 31T^{2} \) |
| 37 | \( 1 + 4.58T + 37T^{2} \) |
| 41 | \( 1 - 2.88T + 41T^{2} \) |
| 43 | \( 1 - 0.0299T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 4.28T + 53T^{2} \) |
| 59 | \( 1 - 9.52T + 59T^{2} \) |
| 67 | \( 1 - 6.52T + 67T^{2} \) |
| 71 | \( 1 + 6.64T + 71T^{2} \) |
| 73 | \( 1 - 1.32T + 73T^{2} \) |
| 79 | \( 1 - 8.08T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 2.28T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−7.990538769935659746009889752016, −7.05580019200504559416244892263, −6.37572306040414924963332085974, −5.59452250189093238876940408779, −5.16254949014422588974777509531, −4.21224868466336976253898673903, −3.37328802933034581310491738411, −2.49455641641998271573229873452, −1.54665663382325354620745414728, −1.03133090111433855269362921970,
1.03133090111433855269362921970, 1.54665663382325354620745414728, 2.49455641641998271573229873452, 3.37328802933034581310491738411, 4.21224868466336976253898673903, 5.16254949014422588974777509531, 5.59452250189093238876940408779, 6.37572306040414924963332085974, 7.05580019200504559416244892263, 7.990538769935659746009889752016
