| L(s) = 1 | − 2-s + 2.72·3-s + 4-s + 5-s − 2.72·6-s + 0.894·7-s − 8-s + 4.44·9-s − 10-s − 5.03·11-s + 2.72·12-s − 5.35·13-s − 0.894·14-s + 2.72·15-s + 16-s + 5.62·17-s − 4.44·18-s − 0.243·19-s + 20-s + 2.44·21-s + 5.03·22-s + 4.21·23-s − 2.72·24-s + 25-s + 5.35·26-s + 3.94·27-s + 0.894·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.57·3-s + 0.5·4-s + 0.447·5-s − 1.11·6-s + 0.338·7-s − 0.353·8-s + 1.48·9-s − 0.316·10-s − 1.51·11-s + 0.787·12-s − 1.48·13-s − 0.239·14-s + 0.704·15-s + 0.250·16-s + 1.36·17-s − 1.04·18-s − 0.0558·19-s + 0.223·20-s + 0.532·21-s + 1.07·22-s + 0.879·23-s − 0.556·24-s + 0.200·25-s + 1.05·26-s + 0.758·27-s + 0.169·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.638166416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.638166416\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 2.72T + 3T^{2} \) |
| 7 | \( 1 - 0.894T + 7T^{2} \) |
| 11 | \( 1 + 5.03T + 11T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 17 | \( 1 - 5.62T + 17T^{2} \) |
| 19 | \( 1 + 0.243T + 19T^{2} \) |
| 23 | \( 1 - 4.21T + 23T^{2} \) |
| 29 | \( 1 - 7.69T + 29T^{2} \) |
| 31 | \( 1 - 1.55T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 3.18T + 41T^{2} \) |
| 43 | \( 1 + 3.19T + 43T^{2} \) |
| 47 | \( 1 - 4.78T + 47T^{2} \) |
| 53 | \( 1 + 5.76T + 53T^{2} \) |
| 59 | \( 1 - 7.35T + 59T^{2} \) |
| 61 | \( 1 - 2.11T + 61T^{2} \) |
| 67 | \( 1 - 14.7T + 67T^{2} \) |
| 71 | \( 1 + 2.41T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 - 8.03T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 1.78T + 97T^{2} \) |
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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
−8.156082931690093092414200044847, −8.009437881216575694815482498628, −7.42736396202558283293059955470, −6.54103043468266146321926230954, −5.31559989367792226567458573713, −4.77331430272373186654861671227, −3.42168187699198144332961927767, −2.56644247691508509384569627846, −2.35508810011104483169060020400, −0.968985558252578111790719246672,
0.968985558252578111790719246672, 2.35508810011104483169060020400, 2.56644247691508509384569627846, 3.42168187699198144332961927767, 4.77331430272373186654861671227, 5.31559989367792226567458573713, 6.54103043468266146321926230954, 7.42736396202558283293059955470, 8.009437881216575694815482498628, 8.156082931690093092414200044847
