| L(s) = 1 | + 2.36·2-s + 3-s + 3.57·4-s + 2.36·6-s + 0.784·7-s + 3.72·8-s + 9-s + 2.93·11-s + 3.57·12-s + 4·13-s + 1.85·14-s + 1.63·16-s − 5.15·17-s + 2.36·18-s − 19-s + 0.784·21-s + 6.93·22-s − 3.78·23-s + 3.72·24-s + 9.44·26-s + 27-s + 2.80·28-s − 5·29-s + 1.06·31-s − 3.57·32-s + 2.93·33-s − 12.1·34-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 0.577·3-s + 1.78·4-s + 0.964·6-s + 0.296·7-s + 1.31·8-s + 0.333·9-s + 0.885·11-s + 1.03·12-s + 1.10·13-s + 0.495·14-s + 0.409·16-s − 1.24·17-s + 0.556·18-s − 0.229·19-s + 0.171·21-s + 1.47·22-s − 0.789·23-s + 0.759·24-s + 1.85·26-s + 0.192·27-s + 0.530·28-s − 0.928·29-s + 0.190·31-s − 0.632·32-s + 0.511·33-s − 2.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.640550080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.640550080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 - 2.36T + 2T^{2} \) |
| 7 | \( 1 - 0.784T + 7T^{2} \) |
| 11 | \( 1 - 2.93T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + 5.15T + 17T^{2} \) |
| 23 | \( 1 + 3.78T + 23T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 1.06T + 31T^{2} \) |
| 37 | \( 1 + 0.722T + 37T^{2} \) |
| 41 | \( 1 - 7.93T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 4.66T + 59T^{2} \) |
| 61 | \( 1 - 8.87T + 61T^{2} \) |
| 67 | \( 1 - 8.93T + 67T^{2} \) |
| 71 | \( 1 + 2.66T + 71T^{2} \) |
| 73 | \( 1 + 8.44T + 73T^{2} \) |
| 79 | \( 1 - 3.35T + 79T^{2} \) |
| 83 | \( 1 - 4.93T + 83T^{2} \) |
| 89 | \( 1 - 0.569T + 89T^{2} \) |
| 97 | \( 1 - 8.59T + 97T^{2} \) |
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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
−9.412369018829414265295174668635, −8.729714797993450808591123651882, −7.78095851205330655340080988979, −6.70833599322353207277584593512, −6.22732384503522644621580693384, −5.25175240876527058877968695226, −4.07028220141468214138204329344, −3.93396579069535076628270207304, −2.65907996937476021672666361868, −1.68212040412198250094898676165,
1.68212040412198250094898676165, 2.65907996937476021672666361868, 3.93396579069535076628270207304, 4.07028220141468214138204329344, 5.25175240876527058877968695226, 6.22732384503522644621580693384, 6.70833599322353207277584593512, 7.78095851205330655340080988979, 8.729714797993450808591123651882, 9.412369018829414265295174668635
