| L(s) = 1 | + 2-s + 4-s − 3.29·5-s + 8-s − 3.29·10-s + 11-s − 6.06·13-s + 16-s − 6.11·17-s − 0.0511·19-s − 3.29·20-s + 22-s + 6.75·23-s + 5.82·25-s − 6.06·26-s − 2.82·29-s − 5.87·31-s + 32-s − 6.11·34-s − 8.31·37-s − 0.0511·38-s − 3.29·40-s + 6.11·41-s + 2.90·43-s + 44-s + 6.75·46-s + 1.22·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.47·5-s + 0.353·8-s − 1.04·10-s + 0.301·11-s − 1.68·13-s + 0.250·16-s − 1.48·17-s − 0.0117·19-s − 0.735·20-s + 0.213·22-s + 1.40·23-s + 1.16·25-s − 1.19·26-s − 0.525·29-s − 1.05·31-s + 0.176·32-s − 1.04·34-s − 1.36·37-s − 0.00830·38-s − 0.520·40-s + 0.955·41-s + 0.442·43-s + 0.150·44-s + 0.996·46-s + 0.178·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.428389843\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.428389843\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 3.29T + 5T^{2} \) |
| 13 | \( 1 + 6.06T + 13T^{2} \) |
| 17 | \( 1 + 6.11T + 17T^{2} \) |
| 19 | \( 1 + 0.0511T + 19T^{2} \) |
| 23 | \( 1 - 6.75T + 23T^{2} \) |
| 29 | \( 1 + 2.82T + 29T^{2} \) |
| 31 | \( 1 + 5.87T + 31T^{2} \) |
| 37 | \( 1 + 8.31T + 37T^{2} \) |
| 41 | \( 1 - 6.11T + 41T^{2} \) |
| 43 | \( 1 - 2.90T + 43T^{2} \) |
| 47 | \( 1 - 1.22T + 47T^{2} \) |
| 53 | \( 1 + 3.00T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 7.23T + 61T^{2} \) |
| 67 | \( 1 + 1.68T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 9.13T + 79T^{2} \) |
| 83 | \( 1 - 0.951T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 - 14.7T + 97T^{2} \) |
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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
−7.35170814516782130737453716772, −7.16798147419579201683405327998, −6.48198029136288287342807993070, −5.33985501034940280663043258970, −4.83893546751801490243629746624, −4.20242521733884771193054980402, −3.59234964998733652369681809273, −2.76327536133862309440637462533, −1.96854263231814288239933533311, −0.48397340880622314036807803054,
0.48397340880622314036807803054, 1.96854263231814288239933533311, 2.76327536133862309440637462533, 3.59234964998733652369681809273, 4.20242521733884771193054980402, 4.83893546751801490243629746624, 5.33985501034940280663043258970, 6.48198029136288287342807993070, 7.16798147419579201683405327998, 7.35170814516782130737453716772
