| L(s) = 1 | − 2.75·2-s + 2.22·3-s + 5.59·4-s + 3.97·5-s − 6.13·6-s + 1.71·7-s − 9.90·8-s + 1.95·9-s − 10.9·10-s + 2.05·11-s + 12.4·12-s − 3.29·13-s − 4.73·14-s + 8.83·15-s + 16.1·16-s + 6.66·17-s − 5.38·18-s − 5.33·19-s + 22.2·20-s + 3.82·21-s − 5.67·22-s + 23-s − 22.0·24-s + 10.7·25-s + 9.07·26-s − 2.33·27-s + 9.61·28-s + ⋯ |
| L(s) = 1 | − 1.94·2-s + 1.28·3-s + 2.79·4-s + 1.77·5-s − 2.50·6-s + 0.649·7-s − 3.50·8-s + 0.650·9-s − 3.46·10-s + 0.620·11-s + 3.59·12-s − 0.913·13-s − 1.26·14-s + 2.28·15-s + 4.02·16-s + 1.61·17-s − 1.26·18-s − 1.22·19-s + 4.96·20-s + 0.834·21-s − 1.20·22-s + 0.208·23-s − 4.50·24-s + 2.15·25-s + 1.78·26-s − 0.448·27-s + 1.81·28-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
| \(L(1)\) |
\(\approx\) |
\(1.412619466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.412619466\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.75T + 2T^{2} \) |
| 3 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 - 3.97T + 5T^{2} \) |
| 7 | \( 1 - 1.71T + 7T^{2} \) |
| 11 | \( 1 - 2.05T + 11T^{2} \) |
| 13 | \( 1 + 3.29T + 13T^{2} \) |
| 17 | \( 1 - 6.66T + 17T^{2} \) |
| 19 | \( 1 + 5.33T + 19T^{2} \) |
| 31 | \( 1 + 7.39T + 31T^{2} \) |
| 37 | \( 1 + 0.411T + 37T^{2} \) |
| 41 | \( 1 + 0.911T + 41T^{2} \) |
| 43 | \( 1 + 8.79T + 43T^{2} \) |
| 47 | \( 1 - 2.09T + 47T^{2} \) |
| 53 | \( 1 - 4.39T + 53T^{2} \) |
| 59 | \( 1 + 10.2T + 59T^{2} \) |
| 61 | \( 1 + 3.66T + 61T^{2} \) |
| 67 | \( 1 - 6.04T + 67T^{2} \) |
| 71 | \( 1 + 3.14T + 71T^{2} \) |
| 73 | \( 1 + 9.21T + 73T^{2} \) |
| 79 | \( 1 - 15.8T + 79T^{2} \) |
| 83 | \( 1 - 4.38T + 83T^{2} \) |
| 89 | \( 1 + 3.34T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−10.08966799893435182052664391281, −9.451619669006417582720142849615, −8.996360212304309957668026955746, −8.189126080738883525255048557341, −7.41517726424255693806043796461, −6.45298791595998468384846901911, −5.44679366377058012267935324936, −3.15600443676083909287705500878, −2.11763972102688109809777699315, −1.57195574705848677486262265441,
1.57195574705848677486262265441, 2.11763972102688109809777699315, 3.15600443676083909287705500878, 5.44679366377058012267935324936, 6.45298791595998468384846901911, 7.41517726424255693806043796461, 8.189126080738883525255048557341, 8.996360212304309957668026955746, 9.451619669006417582720142849615, 10.08966799893435182052664391281
