| L(s) = 1 | − 3·3-s − 2.63·5-s + 7·7-s + 9·9-s − 24.4·11-s − 83.0·13-s + 7.89·15-s + 110.·17-s + 26.7·19-s − 21·21-s + 50.1·23-s − 118.·25-s − 27·27-s − 16.7·29-s + 129.·31-s + 73.2·33-s − 18.4·35-s + 122.·37-s + 249.·39-s − 417.·41-s + 153.·43-s − 23.6·45-s − 175.·47-s + 49·49-s − 330.·51-s − 487.·53-s + 64.3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.235·5-s + 0.377·7-s + 0.333·9-s − 0.669·11-s − 1.77·13-s + 0.135·15-s + 1.57·17-s + 0.322·19-s − 0.218·21-s + 0.454·23-s − 0.944·25-s − 0.192·27-s − 0.107·29-s + 0.748·31-s + 0.386·33-s − 0.0890·35-s + 0.543·37-s + 1.02·39-s − 1.59·41-s + 0.543·43-s − 0.0785·45-s − 0.545·47-s + 0.142·49-s − 0.907·51-s − 1.26·53-s + 0.157·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.274171791\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.274171791\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| good | 5 | \( 1 + 2.63T + 125T^{2} \) |
| 11 | \( 1 + 24.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 83.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 110.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 26.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 50.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 16.7T + 2.43e4T^{2} \) |
| 31 | \( 1 - 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 122.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 417.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 153.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 175.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 487.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 496.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 274.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 396.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 844.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 895.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.31e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.36e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 683.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43e3T + 9.12e5T^{2} \) |
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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
−10.01288080054549075613535605533, −9.589609372386440342928244186054, −8.010620270195945012064933023109, −7.64778218661105754150284479547, −6.57884678652091473885235923400, −5.29357869374435849339575101114, −4.93013828432329278639156850869, −3.49944081804925702356506570638, −2.21765101221328420601629447104, −0.66443723206077433171751991017,
0.66443723206077433171751991017, 2.21765101221328420601629447104, 3.49944081804925702356506570638, 4.93013828432329278639156850869, 5.29357869374435849339575101114, 6.57884678652091473885235923400, 7.64778218661105754150284479547, 8.010620270195945012064933023109, 9.589609372386440342928244186054, 10.01288080054549075613535605533
