L(s) = 1 | − 1.62·2-s − 2.82·3-s + 0.654·4-s + 2.04·5-s + 4.61·6-s − 3.70·7-s + 2.19·8-s + 5.00·9-s − 3.33·10-s + 11-s − 1.85·12-s − 5.86·13-s + 6.04·14-s − 5.78·15-s − 4.88·16-s + 0.842·17-s − 8.16·18-s + 1.22·19-s + 1.33·20-s + 10.4·21-s − 1.62·22-s − 8.32·23-s − 6.20·24-s − 0.817·25-s + 9.56·26-s − 5.68·27-s − 2.42·28-s + ⋯ |
L(s) = 1 | − 1.15·2-s − 1.63·3-s + 0.327·4-s + 0.914·5-s + 1.88·6-s − 1.40·7-s + 0.774·8-s + 1.66·9-s − 1.05·10-s + 0.301·11-s − 0.535·12-s − 1.62·13-s + 1.61·14-s − 1.49·15-s − 1.22·16-s + 0.204·17-s − 1.92·18-s + 0.281·19-s + 0.299·20-s + 2.29·21-s − 0.347·22-s − 1.73·23-s − 1.26·24-s − 0.163·25-s + 1.87·26-s − 1.09·27-s − 0.459·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2797526343\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2797526343\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 61 | \( 1 + T \) |
good | 2 | \( 1 + 1.62T + 2T^{2} \) |
| 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 - 2.04T + 5T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 13 | \( 1 + 5.86T + 13T^{2} \) |
| 17 | \( 1 - 0.842T + 17T^{2} \) |
| 19 | \( 1 - 1.22T + 19T^{2} \) |
| 23 | \( 1 + 8.32T + 23T^{2} \) |
| 29 | \( 1 - 4.26T + 29T^{2} \) |
| 31 | \( 1 + 7.99T + 31T^{2} \) |
| 37 | \( 1 - 2.78T + 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 3.69T + 47T^{2} \) |
| 53 | \( 1 - 3.77T + 53T^{2} \) |
| 59 | \( 1 - 3.44T + 59T^{2} \) |
| 67 | \( 1 + 0.830T + 67T^{2} \) |
| 71 | \( 1 - 7.22T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 - 8.50T + 83T^{2} \) |
| 89 | \( 1 + 7.28T + 89T^{2} \) |
| 97 | \( 1 - 11.5T + 97T^{2} \) |
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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
−10.24765113440015722005243792898, −9.644012965846912018720227691543, −9.412949480873368542209680043759, −7.74593504339846866354200867496, −6.91293900910417092827749109688, −6.11213919618430916399075303029, −5.38550696659829143815949995313, −4.19166692034796888459649412074, −2.20118041319386242090689638366, −0.55959257140641015377214382591,
0.55959257140641015377214382591, 2.20118041319386242090689638366, 4.19166692034796888459649412074, 5.38550696659829143815949995313, 6.11213919618430916399075303029, 6.91293900910417092827749109688, 7.74593504339846866354200867496, 9.412949480873368542209680043759, 9.644012965846912018720227691543, 10.24765113440015722005243792898