| L(s) = 1 | − 2.48·2-s + 3-s + 4.15·4-s − 0.783·5-s − 2.48·6-s + 7-s − 5.33·8-s + 9-s + 1.94·10-s + 4.88·11-s + 4.15·12-s − 6.65·13-s − 2.48·14-s − 0.783·15-s + 4.92·16-s + 1.25·17-s − 2.48·18-s − 2.07·19-s − 3.25·20-s + 21-s − 12.1·22-s + 6.86·23-s − 5.33·24-s − 4.38·25-s + 16.5·26-s + 27-s + 4.15·28-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 0.577·3-s + 2.07·4-s − 0.350·5-s − 1.01·6-s + 0.377·7-s − 1.88·8-s + 0.333·9-s + 0.614·10-s + 1.47·11-s + 1.19·12-s − 1.84·13-s − 0.662·14-s − 0.202·15-s + 1.23·16-s + 0.304·17-s − 0.584·18-s − 0.476·19-s − 0.727·20-s + 0.218·21-s − 2.58·22-s + 1.43·23-s − 1.08·24-s − 0.877·25-s + 3.23·26-s + 0.192·27-s + 0.784·28-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 | \( 1 + 2.48T + 2T^{2} \) |
| 5 | \( 1 + 0.783T + 5T^{2} \) |
| 11 | \( 1 - 4.88T + 11T^{2} \) |
| 13 | \( 1 + 6.65T + 13T^{2} \) |
| 17 | \( 1 - 1.25T + 17T^{2} \) |
| 19 | \( 1 + 2.07T + 19T^{2} \) |
| 23 | \( 1 - 6.86T + 23T^{2} \) |
| 29 | \( 1 + 9.79T + 29T^{2} \) |
| 31 | \( 1 + 3.45T + 31T^{2} \) |
| 37 | \( 1 + 2.67T + 37T^{2} \) |
| 41 | \( 1 + 4.19T + 41T^{2} \) |
| 43 | \( 1 + 4.11T + 43T^{2} \) |
| 47 | \( 1 + 6.51T + 47T^{2} \) |
| 53 | \( 1 - 3.48T + 53T^{2} \) |
| 59 | \( 1 - 12.2T + 59T^{2} \) |
| 61 | \( 1 - 6.13T + 61T^{2} \) |
| 67 | \( 1 + 15.0T + 67T^{2} \) |
| 71 | \( 1 + 7.18T + 71T^{2} \) |
| 73 | \( 1 - 2.24T + 73T^{2} \) |
| 79 | \( 1 + 7.38T + 79T^{2} \) |
| 83 | \( 1 + 3.82T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 16.8T + 97T^{2} \) |
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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
−8.645700365358863118800899306003, −7.81471011334902883939698521431, −7.15835810100353009358038831168, −6.85977045427952598605264969588, −5.50785814672533948360846831379, −4.36235910045696442876869895615, −3.32042580854930806756995235928, −2.18132552610136066356746270128, −1.45554395550009807406699085580, 0,
1.45554395550009807406699085580, 2.18132552610136066356746270128, 3.32042580854930806756995235928, 4.36235910045696442876869895615, 5.50785814672533948360846831379, 6.85977045427952598605264969588, 7.15835810100353009358038831168, 7.81471011334902883939698521431, 8.645700365358863118800899306003
