| L(s) = 1 | − 2.55·2-s + 2.17·3-s + 4.50·4-s + 5-s − 5.54·6-s + 4.37·7-s − 6.38·8-s + 1.72·9-s − 2.55·10-s + 0.414·11-s + 9.78·12-s − 0.0505·13-s − 11.1·14-s + 2.17·15-s + 7.26·16-s + 4.30·17-s − 4.39·18-s + 2.90·19-s + 4.50·20-s + 9.50·21-s − 1.05·22-s + 1.21·23-s − 13.8·24-s + 25-s + 0.128·26-s − 2.77·27-s + 19.6·28-s + ⋯ |
| L(s) = 1 | − 1.80·2-s + 1.25·3-s + 2.25·4-s + 0.447·5-s − 2.26·6-s + 1.65·7-s − 2.25·8-s + 0.574·9-s − 0.806·10-s + 0.124·11-s + 2.82·12-s − 0.0140·13-s − 2.98·14-s + 0.561·15-s + 1.81·16-s + 1.04·17-s − 1.03·18-s + 0.665·19-s + 1.00·20-s + 2.07·21-s − 0.225·22-s + 0.253·23-s − 2.83·24-s + 0.200·25-s + 0.0252·26-s − 0.534·27-s + 3.72·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.125504190\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.125504190\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 1607 | \( 1 + T \) |
| good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 - 2.17T + 3T^{2} \) |
| 7 | \( 1 - 4.37T + 7T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 13 | \( 1 + 0.0505T + 13T^{2} \) |
| 17 | \( 1 - 4.30T + 17T^{2} \) |
| 19 | \( 1 - 2.90T + 19T^{2} \) |
| 23 | \( 1 - 1.21T + 23T^{2} \) |
| 29 | \( 1 + 0.288T + 29T^{2} \) |
| 31 | \( 1 + 9.27T + 31T^{2} \) |
| 37 | \( 1 + 8.86T + 37T^{2} \) |
| 41 | \( 1 - 0.791T + 41T^{2} \) |
| 43 | \( 1 - 8.18T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 - 5.43T + 53T^{2} \) |
| 59 | \( 1 - 3.33T + 59T^{2} \) |
| 61 | \( 1 + 9.22T + 61T^{2} \) |
| 67 | \( 1 - 8.47T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 0.578T + 79T^{2} \) |
| 83 | \( 1 + 13.9T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 0.156T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.015453083100231381793576269747, −7.38068461507937158174504837241, −7.18691036444195634412763346269, −5.79804241703438360873855257377, −5.28446883604623795538654007379, −4.02495703097002465231250434803, −3.09602450543262954189062988707, −2.22869734101277481495723911415, −1.71975237539037766832971139866, −0.949975128034867833038460674342,
0.949975128034867833038460674342, 1.71975237539037766832971139866, 2.22869734101277481495723911415, 3.09602450543262954189062988707, 4.02495703097002465231250434803, 5.28446883604623795538654007379, 5.79804241703438360873855257377, 7.18691036444195634412763346269, 7.38068461507937158174504837241, 8.015453083100231381793576269747
