| L(s) = 1 | + 1.70·2-s − 3-s + 0.914·4-s + 1.08·5-s − 1.70·6-s + 7-s − 1.85·8-s + 9-s + 1.85·10-s + 1.58·11-s − 0.914·12-s − 5.55·13-s + 1.70·14-s − 1.08·15-s − 4.99·16-s − 3.38·17-s + 1.70·18-s − 4.84·19-s + 0.994·20-s − 21-s + 2.69·22-s + 7.35·23-s + 1.85·24-s − 3.81·25-s − 9.48·26-s − 27-s + 0.914·28-s + ⋯ |
| L(s) = 1 | + 1.20·2-s − 0.577·3-s + 0.457·4-s + 0.486·5-s − 0.696·6-s + 0.377·7-s − 0.655·8-s + 0.333·9-s + 0.586·10-s + 0.476·11-s − 0.264·12-s − 1.54·13-s + 0.456·14-s − 0.280·15-s − 1.24·16-s − 0.820·17-s + 0.402·18-s − 1.11·19-s + 0.222·20-s − 0.218·21-s + 0.575·22-s + 1.53·23-s + 0.378·24-s − 0.763·25-s − 1.86·26-s − 0.192·27-s + 0.172·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 - 1.70T + 2T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 + 5.55T + 13T^{2} \) |
| 17 | \( 1 + 3.38T + 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 - 7.35T + 23T^{2} \) |
| 29 | \( 1 - 4.01T + 29T^{2} \) |
| 31 | \( 1 + 8.96T + 31T^{2} \) |
| 37 | \( 1 + 3.94T + 37T^{2} \) |
| 41 | \( 1 - 6.40T + 41T^{2} \) |
| 43 | \( 1 + 8.32T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 2.86T + 53T^{2} \) |
| 59 | \( 1 + 3.19T + 59T^{2} \) |
| 61 | \( 1 - 15.1T + 61T^{2} \) |
| 67 | \( 1 + 14.4T + 67T^{2} \) |
| 71 | \( 1 - 4.60T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 5.60T + 79T^{2} \) |
| 83 | \( 1 + 6.19T + 83T^{2} \) |
| 89 | \( 1 + 14.7T + 89T^{2} \) |
| 97 | \( 1 + 4.99T + 97T^{2} \) |
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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
−8.610613531234914416054330857194, −7.30808331877785910962883951700, −6.73445630021216794639886157800, −5.97782727386622296268829113055, −5.14419454302676225091142268002, −4.71299025550372585868335574914, −3.91599838831241861929857512756, −2.74247301364804884340648600727, −1.84044793250717650033417145737, 0,
1.84044793250717650033417145737, 2.74247301364804884340648600727, 3.91599838831241861929857512756, 4.71299025550372585868335574914, 5.14419454302676225091142268002, 5.97782727386622296268829113055, 6.73445630021216794639886157800, 7.30808331877785910962883951700, 8.610613531234914416054330857194
