| L(s) = 1 | + 2-s + 1.24·3-s + 4-s − 3.80·5-s + 1.24·6-s − 1.10·7-s + 8-s − 1.44·9-s − 3.80·10-s + 0.890·11-s + 1.24·12-s − 1.10·14-s − 4.74·15-s + 16-s + 2.66·17-s − 1.44·18-s + 19-s − 3.80·20-s − 1.38·21-s + 0.890·22-s + 6.31·23-s + 1.24·24-s + 9.45·25-s − 5.54·27-s − 1.10·28-s + 8.09·29-s − 4.74·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.719·3-s + 0.5·4-s − 1.70·5-s + 0.509·6-s − 0.419·7-s + 0.353·8-s − 0.481·9-s − 1.20·10-s + 0.268·11-s + 0.359·12-s − 0.296·14-s − 1.22·15-s + 0.250·16-s + 0.646·17-s − 0.340·18-s + 0.229·19-s − 0.850·20-s − 0.302·21-s + 0.189·22-s + 1.31·23-s + 0.254·24-s + 1.89·25-s − 1.06·27-s − 0.209·28-s + 1.50·29-s − 0.865·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.24T + 3T^{2} \) |
| 5 | \( 1 + 3.80T + 5T^{2} \) |
| 7 | \( 1 + 1.10T + 7T^{2} \) |
| 11 | \( 1 - 0.890T + 11T^{2} \) |
| 17 | \( 1 - 2.66T + 17T^{2} \) |
| 23 | \( 1 - 6.31T + 23T^{2} \) |
| 29 | \( 1 - 8.09T + 29T^{2} \) |
| 31 | \( 1 - 4.85T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 4.98T + 43T^{2} \) |
| 47 | \( 1 + 6.98T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 5.67T + 59T^{2} \) |
| 61 | \( 1 + 1.97T + 61T^{2} \) |
| 67 | \( 1 - 7.74T + 67T^{2} \) |
| 71 | \( 1 - 2.84T + 71T^{2} \) |
| 73 | \( 1 + 2.66T + 73T^{2} \) |
| 79 | \( 1 + 8.36T + 79T^{2} \) |
| 83 | \( 1 + 6.49T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 7.92T + 97T^{2} \) |
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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
−7.71671915509453139952636529500, −6.88001101039620127894526075746, −6.53120493137460340273349512195, −5.21890206732371494428807597721, −4.76744690516469833952062101336, −3.76142181012780093394510280495, −3.22750174915363869338592546521, −2.90770344742900596419423587200, −1.39043080066024993178733485447, 0,
1.39043080066024993178733485447, 2.90770344742900596419423587200, 3.22750174915363869338592546521, 3.76142181012780093394510280495, 4.76744690516469833952062101336, 5.21890206732371494428807597721, 6.53120493137460340273349512195, 6.88001101039620127894526075746, 7.71671915509453139952636529500
