| L(s) = 1 | − 1.28·2-s + 3.25·3-s − 0.338·4-s + 4.18·5-s − 4.19·6-s − 3.67·7-s + 3.01·8-s + 7.59·9-s − 5.39·10-s − 5.13·11-s − 1.10·12-s + 4.32·13-s + 4.74·14-s + 13.6·15-s − 3.20·16-s + 0.381·17-s − 9.78·18-s + 1.30·19-s − 1.41·20-s − 11.9·21-s + 6.62·22-s + 23-s + 9.81·24-s + 12.5·25-s − 5.57·26-s + 14.9·27-s + 1.24·28-s + ⋯ |
| L(s) = 1 | − 0.911·2-s + 1.87·3-s − 0.169·4-s + 1.87·5-s − 1.71·6-s − 1.39·7-s + 1.06·8-s + 2.53·9-s − 1.70·10-s − 1.54·11-s − 0.318·12-s + 1.19·13-s + 1.26·14-s + 3.51·15-s − 0.801·16-s + 0.0924·17-s − 2.30·18-s + 0.300·19-s − 0.317·20-s − 2.61·21-s + 1.41·22-s + 0.208·23-s + 2.00·24-s + 2.50·25-s − 1.09·26-s + 2.87·27-s + 0.235·28-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
| \(L(1)\) |
\(\approx\) |
\(1.942871767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.942871767\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.28T + 2T^{2} \) |
| 3 | \( 1 - 3.25T + 3T^{2} \) |
| 5 | \( 1 - 4.18T + 5T^{2} \) |
| 7 | \( 1 + 3.67T + 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 - 4.32T + 13T^{2} \) |
| 17 | \( 1 - 0.381T + 17T^{2} \) |
| 19 | \( 1 - 1.30T + 19T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 6.52T + 37T^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 + 3.41T + 43T^{2} \) |
| 47 | \( 1 + 4.78T + 47T^{2} \) |
| 53 | \( 1 - 1.50T + 53T^{2} \) |
| 59 | \( 1 - 5.31T + 59T^{2} \) |
| 61 | \( 1 + 7.01T + 61T^{2} \) |
| 67 | \( 1 + 8.40T + 67T^{2} \) |
| 71 | \( 1 + 2.40T + 71T^{2} \) |
| 73 | \( 1 - 1.76T + 73T^{2} \) |
| 79 | \( 1 + 6.13T + 79T^{2} \) |
| 83 | \( 1 - 3.80T + 83T^{2} \) |
| 89 | \( 1 + 1.90T + 89T^{2} \) |
| 97 | \( 1 + 5.23T + 97T^{2} \) |
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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
−10.22721910980649544503845375144, −9.543374377099984454266619256648, −8.892591977983960453905239460929, −8.346942954332322150482425130773, −7.27464744569270606690561897199, −6.30528669816217550303571884595, −5.02829224717660554138854289776, −3.42419672597266797670721294547, −2.60435680906888525694255054089, −1.51752874629228380928264464171,
1.51752874629228380928264464171, 2.60435680906888525694255054089, 3.42419672597266797670721294547, 5.02829224717660554138854289776, 6.30528669816217550303571884595, 7.27464744569270606690561897199, 8.346942954332322150482425130773, 8.892591977983960453905239460929, 9.543374377099984454266619256648, 10.22721910980649544503845375144
