| L(s) = 1 | − 2.59·2-s − 3-s + 4.73·4-s − 1.63·5-s + 2.59·6-s + 7-s − 7.11·8-s + 9-s + 4.23·10-s + 4.79·11-s − 4.73·12-s + 0.263·13-s − 2.59·14-s + 1.63·15-s + 8.98·16-s − 5.12·17-s − 2.59·18-s + 5.93·19-s − 7.73·20-s − 21-s − 12.4·22-s − 4.10·23-s + 7.11·24-s − 2.33·25-s − 0.684·26-s − 27-s + 4.73·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s − 0.577·3-s + 2.36·4-s − 0.730·5-s + 1.05·6-s + 0.377·7-s − 2.51·8-s + 0.333·9-s + 1.34·10-s + 1.44·11-s − 1.36·12-s + 0.0731·13-s − 0.693·14-s + 0.421·15-s + 2.24·16-s − 1.24·17-s − 0.611·18-s + 1.36·19-s − 1.72·20-s − 0.218·21-s − 2.65·22-s − 0.855·23-s + 1.45·24-s − 0.467·25-s − 0.134·26-s − 0.192·27-s + 0.895·28-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 | \( 1 + 2.59T + 2T^{2} \) |
| 5 | \( 1 + 1.63T + 5T^{2} \) |
| 11 | \( 1 - 4.79T + 11T^{2} \) |
| 13 | \( 1 - 0.263T + 13T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 5.93T + 19T^{2} \) |
| 23 | \( 1 + 4.10T + 23T^{2} \) |
| 29 | \( 1 - 0.459T + 29T^{2} \) |
| 31 | \( 1 + 9.24T + 31T^{2} \) |
| 37 | \( 1 - 6.64T + 37T^{2} \) |
| 41 | \( 1 + 3.43T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 + 5.69T + 47T^{2} \) |
| 53 | \( 1 - 2.55T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 1.34T + 67T^{2} \) |
| 71 | \( 1 - 5.05T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 0.111T + 79T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 - 11.2T + 89T^{2} \) |
| 97 | \( 1 + 5.51T + 97T^{2} \) |
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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
−8.523564191631556482262855701639, −7.78267069471092795811538652296, −7.18905123709636673843362451129, −6.53009813169340474051049480848, −5.73341648011973911499580796106, −4.41481029934619132925097402833, −3.50596462357727419584450694618, −2.05378133306747786476582478924, −1.18485713307862395004252992652, 0,
1.18485713307862395004252992652, 2.05378133306747786476582478924, 3.50596462357727419584450694618, 4.41481029934619132925097402833, 5.73341648011973911499580796106, 6.53009813169340474051049480848, 7.18905123709636673843362451129, 7.78267069471092795811538652296, 8.523564191631556482262855701639
