| L(s) = 1 | + 2.27·2-s − 1.70·3-s + 3.17·4-s + 5-s − 3.87·6-s − 2.70·7-s + 2.67·8-s − 0.100·9-s + 2.27·10-s + 2.17·11-s − 5.40·12-s + 5.37·13-s − 6.15·14-s − 1.70·15-s − 0.261·16-s − 0.229·18-s + 8.53·19-s + 3.17·20-s + 4.60·21-s + 4.95·22-s + 5.32·23-s − 4.55·24-s + 25-s + 12.2·26-s + 5.27·27-s − 8.60·28-s − 1.31·29-s + ⋯ |
| L(s) = 1 | + 1.60·2-s − 0.983·3-s + 1.58·4-s + 0.447·5-s − 1.58·6-s − 1.02·7-s + 0.946·8-s − 0.0335·9-s + 0.719·10-s + 0.656·11-s − 1.56·12-s + 1.49·13-s − 1.64·14-s − 0.439·15-s − 0.0654·16-s − 0.0540·18-s + 1.95·19-s + 0.710·20-s + 1.00·21-s + 1.05·22-s + 1.11·23-s − 0.930·24-s + 0.200·25-s + 2.40·26-s + 1.01·27-s − 1.62·28-s − 0.243·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.188167804\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.188167804\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.27T + 2T^{2} \) |
| 3 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 + 2.70T + 7T^{2} \) |
| 11 | \( 1 - 2.17T + 11T^{2} \) |
| 13 | \( 1 - 5.37T + 13T^{2} \) |
| 19 | \( 1 - 8.53T + 19T^{2} \) |
| 23 | \( 1 - 5.32T + 23T^{2} \) |
| 29 | \( 1 + 1.31T + 29T^{2} \) |
| 31 | \( 1 + 0.757T + 31T^{2} \) |
| 37 | \( 1 - 8.20T + 37T^{2} \) |
| 41 | \( 1 + 0.0130T + 41T^{2} \) |
| 43 | \( 1 + 0.330T + 43T^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + 8.04T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 + 7.20T + 61T^{2} \) |
| 67 | \( 1 - 2.67T + 67T^{2} \) |
| 71 | \( 1 - 1.87T + 71T^{2} \) |
| 73 | \( 1 + 11.9T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 3.35T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−9.546843939737215936284167469069, −8.969549569741523053252082252157, −7.41432516054121474599264069114, −6.48156571119176251807700335225, −6.06938633232238551725542552646, −5.50864712624351019144928370870, −4.60803494644464968157204130111, −3.49590286052289538889790744955, −2.94141547066010967396959781277, −1.14066174780778228871469816866,
1.14066174780778228871469816866, 2.94141547066010967396959781277, 3.49590286052289538889790744955, 4.60803494644464968157204130111, 5.50864712624351019144928370870, 6.06938633232238551725542552646, 6.48156571119176251807700335225, 7.41432516054121474599264069114, 8.969549569741523053252082252157, 9.546843939737215936284167469069
