| L(s) = 1 | − 2.39·2-s − 3-s + 3.72·4-s + 0.325·5-s + 2.39·6-s − 0.363·7-s − 4.12·8-s + 9-s − 0.777·10-s − 0.861·11-s − 3.72·12-s − 0.164·13-s + 0.868·14-s − 0.325·15-s + 2.41·16-s − 17-s − 2.39·18-s − 8.07·19-s + 1.21·20-s + 0.363·21-s + 2.05·22-s + 3.99·23-s + 4.12·24-s − 4.89·25-s + 0.392·26-s − 27-s − 1.35·28-s + ⋯ |
| L(s) = 1 | − 1.69·2-s − 0.577·3-s + 1.86·4-s + 0.145·5-s + 0.976·6-s − 0.137·7-s − 1.45·8-s + 0.333·9-s − 0.245·10-s − 0.259·11-s − 1.07·12-s − 0.0455·13-s + 0.232·14-s − 0.0839·15-s + 0.603·16-s − 0.242·17-s − 0.563·18-s − 1.85·19-s + 0.270·20-s + 0.0792·21-s + 0.439·22-s + 0.833·23-s + 0.841·24-s − 0.978·25-s + 0.0769·26-s − 0.192·27-s − 0.255·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 157 | \( 1 + T \) |
| good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 5 | \( 1 - 0.325T + 5T^{2} \) |
| 7 | \( 1 + 0.363T + 7T^{2} \) |
| 11 | \( 1 + 0.861T + 11T^{2} \) |
| 13 | \( 1 + 0.164T + 13T^{2} \) |
| 19 | \( 1 + 8.07T + 19T^{2} \) |
| 23 | \( 1 - 3.99T + 23T^{2} \) |
| 29 | \( 1 + 3.43T + 29T^{2} \) |
| 31 | \( 1 - 5.64T + 31T^{2} \) |
| 37 | \( 1 - 5.74T + 37T^{2} \) |
| 41 | \( 1 + 3.17T + 41T^{2} \) |
| 43 | \( 1 - 8.78T + 43T^{2} \) |
| 47 | \( 1 - 4.83T + 47T^{2} \) |
| 53 | \( 1 + 4.72T + 53T^{2} \) |
| 59 | \( 1 - 5.82T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 12.9T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 - 5.13T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 - 17.9T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
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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
−7.72647674046584063887432149581, −6.80732487566116751250415261391, −6.45039282279772654743456168869, −5.70349217216023980840267319407, −4.71489447521073358905397040470, −3.90654106281295940952722034004, −2.58573161718171439660995881078, −2.00851800100680957001713587430, −0.927247716246292933203885636735, 0,
0.927247716246292933203885636735, 2.00851800100680957001713587430, 2.58573161718171439660995881078, 3.90654106281295940952722034004, 4.71489447521073358905397040470, 5.70349217216023980840267319407, 6.45039282279772654743456168869, 6.80732487566116751250415261391, 7.72647674046584063887432149581
