L(s) = 1 | + 2-s − 2.49·3-s + 4-s + 2.01·5-s − 2.49·6-s + 2.26·7-s + 8-s + 3.22·9-s + 2.01·10-s − 3.70·11-s − 2.49·12-s − 3.55·13-s + 2.26·14-s − 5.02·15-s + 16-s + 6.88·17-s + 3.22·18-s + 2.37·19-s + 2.01·20-s − 5.64·21-s − 3.70·22-s − 2.90·23-s − 2.49·24-s − 0.943·25-s − 3.55·26-s − 0.550·27-s + 2.26·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.43·3-s + 0.5·4-s + 0.900·5-s − 1.01·6-s + 0.854·7-s + 0.353·8-s + 1.07·9-s + 0.636·10-s − 1.11·11-s − 0.719·12-s − 0.986·13-s + 0.604·14-s − 1.29·15-s + 0.250·16-s + 1.67·17-s + 0.759·18-s + 0.544·19-s + 0.450·20-s − 1.23·21-s − 0.789·22-s − 0.605·23-s − 0.509·24-s − 0.188·25-s − 0.697·26-s − 0.105·27-s + 0.427·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.255889718\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.255889718\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2017 | \( 1 + T \) |
good | 3 | \( 1 + 2.49T + 3T^{2} \) |
| 5 | \( 1 - 2.01T + 5T^{2} \) |
| 7 | \( 1 - 2.26T + 7T^{2} \) |
| 11 | \( 1 + 3.70T + 11T^{2} \) |
| 13 | \( 1 + 3.55T + 13T^{2} \) |
| 17 | \( 1 - 6.88T + 17T^{2} \) |
| 19 | \( 1 - 2.37T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 - 5.51T + 29T^{2} \) |
| 31 | \( 1 + 6.05T + 31T^{2} \) |
| 37 | \( 1 - 4.05T + 37T^{2} \) |
| 41 | \( 1 - 5.33T + 41T^{2} \) |
| 43 | \( 1 + 1.25T + 43T^{2} \) |
| 47 | \( 1 - 6.32T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 4.09T + 59T^{2} \) |
| 61 | \( 1 - 2.55T + 61T^{2} \) |
| 67 | \( 1 - 15.7T + 67T^{2} \) |
| 71 | \( 1 - 0.641T + 71T^{2} \) |
| 73 | \( 1 + 8.54T + 73T^{2} \) |
| 79 | \( 1 - 2.47T + 79T^{2} \) |
| 83 | \( 1 - 5.59T + 83T^{2} \) |
| 89 | \( 1 - 8.40T + 89T^{2} \) |
| 97 | \( 1 + 7.21T + 97T^{2} \) |
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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
−8.081373583328741544523762563419, −7.58586557871146526444373879796, −6.77899870165873952712527935534, −5.75850274192906960059261281380, −5.49648246256112941221517227979, −5.07069563878604470996903303313, −4.18735012499167105992756961568, −2.86883529338542702055814518694, −1.98270715755214672198979568139, −0.844478867432388599499347914190,
0.844478867432388599499347914190, 1.98270715755214672198979568139, 2.86883529338542702055814518694, 4.18735012499167105992756961568, 5.07069563878604470996903303313, 5.49648246256112941221517227979, 5.75850274192906960059261281380, 6.77899870165873952712527935534, 7.58586557871146526444373879796, 8.081373583328741544523762563419