| L(s) = 1 | + 2.51·2-s + 4.32·4-s + 0.193·5-s + 3.32·7-s + 5.83·8-s + 0.485·10-s − 11-s − 13-s + 8.34·14-s + 6.02·16-s − 1.51·17-s + 0.679·19-s + 0.835·20-s − 2.51·22-s + 5.70·23-s − 4.96·25-s − 2.51·26-s + 14.3·28-s − 7.12·29-s + 5.80·31-s + 3.48·32-s − 3.80·34-s + 0.641·35-s + 3.41·37-s + 1.70·38-s + 1.12·40-s − 11.3·41-s + ⋯ |
| L(s) = 1 | + 1.77·2-s + 2.16·4-s + 0.0864·5-s + 1.25·7-s + 2.06·8-s + 0.153·10-s − 0.301·11-s − 0.277·13-s + 2.23·14-s + 1.50·16-s − 0.367·17-s + 0.155·19-s + 0.186·20-s − 0.536·22-s + 1.19·23-s − 0.992·25-s − 0.493·26-s + 2.71·28-s − 1.32·29-s + 1.04·31-s + 0.616·32-s − 0.652·34-s + 0.108·35-s + 0.561·37-s + 0.276·38-s + 0.178·40-s − 1.77·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.314714070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.314714070\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 2.51T + 2T^{2} \) |
| 5 | \( 1 - 0.193T + 5T^{2} \) |
| 7 | \( 1 - 3.32T + 7T^{2} \) |
| 17 | \( 1 + 1.51T + 17T^{2} \) |
| 19 | \( 1 - 0.679T + 19T^{2} \) |
| 23 | \( 1 - 5.70T + 23T^{2} \) |
| 29 | \( 1 + 7.12T + 29T^{2} \) |
| 31 | \( 1 - 5.80T + 31T^{2} \) |
| 37 | \( 1 - 3.41T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 - 2.15T + 43T^{2} \) |
| 47 | \( 1 - 1.61T + 47T^{2} \) |
| 53 | \( 1 + 1.02T + 53T^{2} \) |
| 59 | \( 1 + 4.38T + 59T^{2} \) |
| 61 | \( 1 - 7.02T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 4.64T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 - 4.48T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 12.4T + 89T^{2} \) |
| 97 | \( 1 + 7.41T + 97T^{2} \) |
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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
−9.900997325989948141653102467676, −8.678019524205251145874681744165, −7.71659248247109091114393745885, −7.03415028678023849753990227545, −6.02462022612788536757945624415, −5.19826743288126201525072971381, −4.69027228248680363232595250781, −3.75985174227061206685637081289, −2.65235514425428752996197963373, −1.69281072522119708043132529546,
1.69281072522119708043132529546, 2.65235514425428752996197963373, 3.75985174227061206685637081289, 4.69027228248680363232595250781, 5.19826743288126201525072971381, 6.02462022612788536757945624415, 7.03415028678023849753990227545, 7.71659248247109091114393745885, 8.678019524205251145874681744165, 9.900997325989948141653102467676
