| L(s) = 1 | + 1.54·3-s + 5-s − 2.64·7-s − 0.604·9-s + 5.27·11-s + 5.02·13-s + 1.54·15-s + 7.44·17-s + 2.84·19-s − 4.08·21-s − 1.30·23-s + 25-s − 5.57·27-s + 6.30·29-s + 2.75·31-s + 8.16·33-s − 2.64·35-s − 5.12·37-s + 7.76·39-s − 6.47·41-s − 6.46·43-s − 0.604·45-s + 11.6·47-s − 0.0202·49-s + 11.5·51-s − 7.45·53-s + 5.27·55-s + ⋯ |
| L(s) = 1 | + 0.893·3-s + 0.447·5-s − 0.998·7-s − 0.201·9-s + 1.59·11-s + 1.39·13-s + 0.399·15-s + 1.80·17-s + 0.653·19-s − 0.892·21-s − 0.272·23-s + 0.200·25-s − 1.07·27-s + 1.17·29-s + 0.494·31-s + 1.42·33-s − 0.446·35-s − 0.841·37-s + 1.24·39-s − 1.01·41-s − 0.986·43-s − 0.0901·45-s + 1.69·47-s − 0.00289·49-s + 1.61·51-s − 1.02·53-s + 0.711·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.572083379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.572083379\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
| good | 3 | \( 1 - 1.54T + 3T^{2} \) |
| 7 | \( 1 + 2.64T + 7T^{2} \) |
| 11 | \( 1 - 5.27T + 11T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 - 7.44T + 17T^{2} \) |
| 19 | \( 1 - 2.84T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 - 6.30T + 29T^{2} \) |
| 31 | \( 1 - 2.75T + 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 6.46T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 7.45T + 53T^{2} \) |
| 59 | \( 1 + 15.3T + 59T^{2} \) |
| 61 | \( 1 - 2.33T + 61T^{2} \) |
| 67 | \( 1 - 8.20T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 - 7.96T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 2.74T + 83T^{2} \) |
| 89 | \( 1 - 8.81T + 89T^{2} \) |
| 97 | \( 1 - 7.80T + 97T^{2} \) |
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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
−8.047846004586179862446677085114, −7.07770906645526602784975999239, −6.35150645892509500508142360470, −6.00370420856270512269212782351, −5.11401679076386349289591036658, −3.87865253538358334212313935045, −3.40782619624563373997527926506, −2.98267073962409053536740446712, −1.70365220692637768444049751592, −0.970576849488547071182094048960,
0.970576849488547071182094048960, 1.70365220692637768444049751592, 2.98267073962409053536740446712, 3.40782619624563373997527926506, 3.87865253538358334212313935045, 5.11401679076386349289591036658, 6.00370420856270512269212782351, 6.35150645892509500508142360470, 7.07770906645526602784975999239, 8.047846004586179862446677085114
