| L(s) = 1 | − 3-s + 0.408·5-s + 1.71·7-s + 9-s − 4.20·11-s − 0.408·15-s + 5.92·17-s + 0.457·19-s − 1.71·21-s + 2.66·23-s − 4.83·25-s − 27-s − 0.393·29-s − 5.77·31-s + 4.20·33-s + 0.701·35-s + 6.31·37-s + 5.39·41-s + 3.71·43-s + 0.408·45-s + 6.07·47-s − 4.04·49-s − 5.92·51-s − 1.69·53-s − 1.71·55-s − 0.457·57-s − 6.69·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.182·5-s + 0.649·7-s + 0.333·9-s − 1.26·11-s − 0.105·15-s + 1.43·17-s + 0.104·19-s − 0.375·21-s + 0.556·23-s − 0.966·25-s − 0.192·27-s − 0.0731·29-s − 1.03·31-s + 0.732·33-s + 0.118·35-s + 1.03·37-s + 0.842·41-s + 0.566·43-s + 0.0608·45-s + 0.885·47-s − 0.577·49-s − 0.830·51-s − 0.233·53-s − 0.231·55-s − 0.0605·57-s − 0.871·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593600789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593600789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 0.408T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 + 4.20T + 11T^{2} \) |
| 17 | \( 1 - 5.92T + 17T^{2} \) |
| 19 | \( 1 - 0.457T + 19T^{2} \) |
| 23 | \( 1 - 2.66T + 23T^{2} \) |
| 29 | \( 1 + 0.393T + 29T^{2} \) |
| 31 | \( 1 + 5.77T + 31T^{2} \) |
| 37 | \( 1 - 6.31T + 37T^{2} \) |
| 41 | \( 1 - 5.39T + 41T^{2} \) |
| 43 | \( 1 - 3.71T + 43T^{2} \) |
| 47 | \( 1 - 6.07T + 47T^{2} \) |
| 53 | \( 1 + 1.69T + 53T^{2} \) |
| 59 | \( 1 + 6.69T + 59T^{2} \) |
| 61 | \( 1 - 9.84T + 61T^{2} \) |
| 67 | \( 1 - 10.2T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 4.58T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 + 3.97T + 89T^{2} \) |
| 97 | \( 1 - 0.852T + 97T^{2} \) |
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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
−8.175289840742380468662222475565, −7.73663989064043965043498191406, −7.12334863759708478524488453331, −5.93559969706173669569075907086, −5.52823239874079120072419638327, −4.87179085948096073317753387735, −3.93156765471963610603419181678, −2.89370740211512271968668767918, −1.89912466543744867468226374615, −0.75136488953602182380572844961,
0.75136488953602182380572844961, 1.89912466543744867468226374615, 2.89370740211512271968668767918, 3.93156765471963610603419181678, 4.87179085948096073317753387735, 5.52823239874079120072419638327, 5.93559969706173669569075907086, 7.12334863759708478524488453331, 7.73663989064043965043498191406, 8.175289840742380468662222475565
