| L(s) = 1 | + 1.26·2-s − 3-s − 0.401·4-s + 1.64·5-s − 1.26·6-s + 5.15·7-s − 3.03·8-s + 9-s + 2.07·10-s − 5.89·11-s + 0.401·12-s + 2.01·13-s + 6.51·14-s − 1.64·15-s − 3.03·16-s − 17-s + 1.26·18-s + 1.52·19-s − 0.658·20-s − 5.15·21-s − 7.44·22-s + 0.245·23-s + 3.03·24-s − 2.30·25-s + 2.54·26-s − 27-s − 2.06·28-s + ⋯ |
| L(s) = 1 | + 0.894·2-s − 0.577·3-s − 0.200·4-s + 0.733·5-s − 0.516·6-s + 1.94·7-s − 1.07·8-s + 0.333·9-s + 0.655·10-s − 1.77·11-s + 0.115·12-s + 0.558·13-s + 1.74·14-s − 0.423·15-s − 0.759·16-s − 0.242·17-s + 0.298·18-s + 0.350·19-s − 0.147·20-s − 1.12·21-s − 1.58·22-s + 0.0512·23-s + 0.619·24-s − 0.461·25-s + 0.499·26-s − 0.192·27-s − 0.391·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.851102255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.851102255\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 1.26T + 2T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 - 5.15T + 7T^{2} \) |
| 11 | \( 1 + 5.89T + 11T^{2} \) |
| 13 | \( 1 - 2.01T + 13T^{2} \) |
| 19 | \( 1 - 1.52T + 19T^{2} \) |
| 23 | \( 1 - 0.245T + 23T^{2} \) |
| 29 | \( 1 - 9.74T + 29T^{2} \) |
| 31 | \( 1 - 7.83T + 31T^{2} \) |
| 37 | \( 1 - 3.04T + 37T^{2} \) |
| 41 | \( 1 + 6.30T + 41T^{2} \) |
| 43 | \( 1 - 2.84T + 43T^{2} \) |
| 47 | \( 1 - 5.05T + 47T^{2} \) |
| 53 | \( 1 + 7.85T + 53T^{2} \) |
| 59 | \( 1 - 5.24T + 59T^{2} \) |
| 61 | \( 1 - 0.259T + 61T^{2} \) |
| 67 | \( 1 - 2.83T + 67T^{2} \) |
| 71 | \( 1 + 1.86T + 71T^{2} \) |
| 73 | \( 1 + 0.762T + 73T^{2} \) |
| 83 | \( 1 + 0.429T + 83T^{2} \) |
| 89 | \( 1 - 13.0T + 89T^{2} \) |
| 97 | \( 1 - 13.5T + 97T^{2} \) |
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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
−8.159196133847917721846137098162, −7.972199409073044966746263018160, −6.71636261160567581993177527950, −5.84332382882555214220308449039, −5.33470764556001137402572119061, −4.79542244486411584420524481065, −4.30249888379734053561617294924, −2.92386816310451207381751858155, −2.11309889278927445416723732994, −0.898969197176852867278467789047,
0.898969197176852867278467789047, 2.11309889278927445416723732994, 2.92386816310451207381751858155, 4.30249888379734053561617294924, 4.79542244486411584420524481065, 5.33470764556001137402572119061, 5.84332382882555214220308449039, 6.71636261160567581993177527950, 7.972199409073044966746263018160, 8.159196133847917721846137098162
