| L(s) = 1 | + 2.14·2-s − 1.74·3-s + 2.61·4-s − 1.42·5-s − 3.73·6-s + 3.61·7-s + 1.32·8-s + 0.0298·9-s − 3.06·10-s + 4.28·11-s − 4.55·12-s + 0.939·13-s + 7.77·14-s + 2.47·15-s − 2.38·16-s + 7.48·17-s + 0.0640·18-s + 4.97·19-s − 3.72·20-s − 6.29·21-s + 9.21·22-s + 23-s − 2.30·24-s − 2.97·25-s + 2.01·26-s + 5.17·27-s + 9.46·28-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 1.00·3-s + 1.30·4-s − 0.637·5-s − 1.52·6-s + 1.36·7-s + 0.467·8-s + 0.00993·9-s − 0.967·10-s + 1.29·11-s − 1.31·12-s + 0.260·13-s + 2.07·14-s + 0.640·15-s − 0.597·16-s + 1.81·17-s + 0.0150·18-s + 1.14·19-s − 0.833·20-s − 1.37·21-s + 1.96·22-s + 0.208·23-s − 0.470·24-s − 0.594·25-s + 0.395·26-s + 0.994·27-s + 1.78·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.623545047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.623545047\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 - 2.14T + 2T^{2} \) |
| 3 | \( 1 + 1.74T + 3T^{2} \) |
| 5 | \( 1 + 1.42T + 5T^{2} \) |
| 7 | \( 1 - 3.61T + 7T^{2} \) |
| 11 | \( 1 - 4.28T + 11T^{2} \) |
| 13 | \( 1 - 0.939T + 13T^{2} \) |
| 17 | \( 1 - 7.48T + 17T^{2} \) |
| 19 | \( 1 - 4.97T + 19T^{2} \) |
| 31 | \( 1 + 2.22T + 31T^{2} \) |
| 37 | \( 1 + 0.191T + 37T^{2} \) |
| 41 | \( 1 + 5.36T + 41T^{2} \) |
| 43 | \( 1 + 6.26T + 43T^{2} \) |
| 47 | \( 1 - 0.471T + 47T^{2} \) |
| 53 | \( 1 + 6.09T + 53T^{2} \) |
| 59 | \( 1 - 7.35T + 59T^{2} \) |
| 61 | \( 1 - 9.04T + 61T^{2} \) |
| 67 | \( 1 - 8.32T + 67T^{2} \) |
| 71 | \( 1 - 4.71T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 7.80T + 79T^{2} \) |
| 83 | \( 1 + 16.6T + 83T^{2} \) |
| 89 | \( 1 - 2.37T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−11.24313371987429849249774312729, −9.939767503215517344766178930303, −8.597744832618334875356684780183, −7.59961670560775119564500124720, −6.64186608092619527813632525861, −5.55191483351497135851655182607, −5.20023742327574309789013750377, −4.12818566240436072984714480343, −3.31017654307935347582321094977, −1.36375103132713772730030699039,
1.36375103132713772730030699039, 3.31017654307935347582321094977, 4.12818566240436072984714480343, 5.20023742327574309789013750377, 5.55191483351497135851655182607, 6.64186608092619527813632525861, 7.59961670560775119564500124720, 8.597744832618334875356684780183, 9.939767503215517344766178930303, 11.24313371987429849249774312729
