| L(s) = 1 | − 0.208·2-s − 2.00·3-s − 1.95·4-s − 2.17·5-s + 0.418·6-s − 2.92·7-s + 0.826·8-s + 1.01·9-s + 0.453·10-s − 4.59·11-s + 3.92·12-s − 4.31·13-s + 0.611·14-s + 4.35·15-s + 3.74·16-s − 1.26·17-s − 0.212·18-s + 2.45·19-s + 4.25·20-s + 5.86·21-s + 0.960·22-s + 23-s − 1.65·24-s − 0.279·25-s + 0.901·26-s + 3.97·27-s + 5.72·28-s + ⋯ |
| L(s) = 1 | − 0.147·2-s − 1.15·3-s − 0.978·4-s − 0.971·5-s + 0.170·6-s − 1.10·7-s + 0.292·8-s + 0.338·9-s + 0.143·10-s − 1.38·11-s + 1.13·12-s − 1.19·13-s + 0.163·14-s + 1.12·15-s + 0.935·16-s − 0.306·17-s − 0.0499·18-s + 0.563·19-s + 0.950·20-s + 1.28·21-s + 0.204·22-s + 0.208·23-s − 0.338·24-s − 0.0558·25-s + 0.176·26-s + 0.765·27-s + 1.08·28-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
| \(L(1)\) |
\(\approx\) |
\(0.09616257281\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09616257281\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.208T + 2T^{2} \) |
| 3 | \( 1 + 2.00T + 3T^{2} \) |
| 5 | \( 1 + 2.17T + 5T^{2} \) |
| 7 | \( 1 + 2.92T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 + 4.31T + 13T^{2} \) |
| 17 | \( 1 + 1.26T + 17T^{2} \) |
| 19 | \( 1 - 2.45T + 19T^{2} \) |
| 31 | \( 1 + 9.32T + 31T^{2} \) |
| 37 | \( 1 - 0.827T + 37T^{2} \) |
| 41 | \( 1 - 1.17T + 41T^{2} \) |
| 43 | \( 1 - 2.72T + 43T^{2} \) |
| 47 | \( 1 - 2.66T + 47T^{2} \) |
| 53 | \( 1 - 6.56T + 53T^{2} \) |
| 59 | \( 1 + 6.48T + 59T^{2} \) |
| 61 | \( 1 - 9.91T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 12.5T + 73T^{2} \) |
| 79 | \( 1 + 10.5T + 79T^{2} \) |
| 83 | \( 1 - 0.738T + 83T^{2} \) |
| 89 | \( 1 + 13.5T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−10.47852148802038236251228857737, −9.797106474926894916235938361718, −8.885109779967284646937408351342, −7.73591134273462269206210892916, −7.13881145463781426451812790202, −5.75062189938216686934191485498, −5.14852012094043953724194937765, −4.15457602586821943368758882446, −2.95438367279366881559048309581, −0.26837458967246695331760195596,
0.26837458967246695331760195596, 2.95438367279366881559048309581, 4.15457602586821943368758882446, 5.14852012094043953724194937765, 5.75062189938216686934191485498, 7.13881145463781426451812790202, 7.73591134273462269206210892916, 8.885109779967284646937408351342, 9.797106474926894916235938361718, 10.47852148802038236251228857737
