| L(s) = 1 | − 1.43·2-s + 2.46·3-s + 0.0548·4-s − 3.53·6-s + 5.17·7-s + 2.78·8-s + 3.07·9-s + 2.79·11-s + 0.135·12-s − 1.04·13-s − 7.42·14-s − 4.10·16-s − 1.22·17-s − 4.40·18-s − 0.657·19-s + 12.7·21-s − 4.00·22-s − 5.23·23-s + 6.87·24-s + 1.49·26-s + 0.178·27-s + 0.284·28-s + 2.90·29-s − 0.0888·31-s + 0.310·32-s + 6.87·33-s + 1.76·34-s + ⋯ |
| L(s) = 1 | − 1.01·2-s + 1.42·3-s + 0.0274·4-s − 1.44·6-s + 1.95·7-s + 0.985·8-s + 1.02·9-s + 0.841·11-s + 0.0390·12-s − 0.289·13-s − 1.98·14-s − 1.02·16-s − 0.297·17-s − 1.03·18-s − 0.150·19-s + 2.78·21-s − 0.853·22-s − 1.09·23-s + 1.40·24-s + 0.293·26-s + 0.0343·27-s + 0.0537·28-s + 0.540·29-s − 0.0159·31-s + 0.0548·32-s + 1.19·33-s + 0.301·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.509637809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.509637809\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 + 1.43T + 2T^{2} \) |
| 3 | \( 1 - 2.46T + 3T^{2} \) |
| 7 | \( 1 - 5.17T + 7T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 + 1.04T + 13T^{2} \) |
| 17 | \( 1 + 1.22T + 17T^{2} \) |
| 19 | \( 1 + 0.657T + 19T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 - 2.90T + 29T^{2} \) |
| 31 | \( 1 + 0.0888T + 31T^{2} \) |
| 37 | \( 1 - 8.62T + 37T^{2} \) |
| 41 | \( 1 - 0.475T + 41T^{2} \) |
| 43 | \( 1 + 9.62T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 + 4.33T + 53T^{2} \) |
| 59 | \( 1 - 6.20T + 59T^{2} \) |
| 61 | \( 1 + 2.05T + 61T^{2} \) |
| 67 | \( 1 - 9.29T + 67T^{2} \) |
| 71 | \( 1 - 1.61T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 0.552T + 79T^{2} \) |
| 83 | \( 1 - 2.67T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 7.94T + 97T^{2} \) |
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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
−8.204738201419833931051254268818, −7.70397291081926266839568710356, −7.24978854179083968826065011530, −6.07839924723905459818112916456, −4.88769502677974297348446743706, −4.38638443891726267910922437323, −3.69097060326723887479979352432, −2.32295019700265536009463696272, −1.86995682855007318964998813191, −0.977041634522001489283001728517,
0.977041634522001489283001728517, 1.86995682855007318964998813191, 2.32295019700265536009463696272, 3.69097060326723887479979352432, 4.38638443891726267910922437323, 4.88769502677974297348446743706, 6.07839924723905459818112916456, 7.24978854179083968826065011530, 7.70397291081926266839568710356, 8.204738201419833931051254268818
