| L(s) = 1 | + 2-s + 4-s + 1.41·5-s + 8-s + 1.41·10-s + 4·11-s + 4.24·13-s + 16-s − 7.07·17-s − 5.65·19-s + 1.41·20-s + 4·22-s + 8·23-s − 2.99·25-s + 4.24·26-s + 2·29-s + 32-s − 7.07·34-s + 4·37-s − 5.65·38-s + 1.41·40-s + 9.89·41-s − 4·43-s + 4·44-s + 8·46-s + 5.65·47-s − 2.99·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.632·5-s + 0.353·8-s + 0.447·10-s + 1.20·11-s + 1.17·13-s + 0.250·16-s − 1.71·17-s − 1.29·19-s + 0.316·20-s + 0.852·22-s + 1.66·23-s − 0.599·25-s + 0.832·26-s + 0.371·29-s + 0.176·32-s − 1.21·34-s + 0.657·37-s − 0.917·38-s + 0.223·40-s + 1.54·41-s − 0.609·43-s + 0.603·44-s + 1.17·46-s + 0.825·47-s − 0.424·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.859230500\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.859230500\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 + 5.65T + 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−10.35138578049123002607309059887, −8.986351798200098119872455168136, −8.835374543565323964485863950256, −7.30771536769475545877594253009, −6.33446421236280350188296302066, −6.06068699843142175191593109764, −4.61622541306224796762577273579, −3.97821097725850109395275477104, −2.64428853613751625658277650408, −1.46445956865435966907113938437,
1.46445956865435966907113938437, 2.64428853613751625658277650408, 3.97821097725850109395275477104, 4.61622541306224796762577273579, 6.06068699843142175191593109764, 6.33446421236280350188296302066, 7.30771536769475545877594253009, 8.835374543565323964485863950256, 8.986351798200098119872455168136, 10.35138578049123002607309059887
