| L(s) = 1 | − 1.41·2-s − 1.33·3-s + 0.0159·4-s − 3.46·5-s + 1.89·6-s − 4.25·7-s + 2.81·8-s − 1.22·9-s + 4.92·10-s + 11-s − 0.0212·12-s + 4.11·13-s + 6.04·14-s + 4.61·15-s − 4.03·16-s + 4.97·17-s + 1.74·18-s − 4.15·19-s − 0.0554·20-s + 5.66·21-s − 1.41·22-s − 3.75·24-s + 7.02·25-s − 5.83·26-s + 5.62·27-s − 0.0680·28-s − 4.41·29-s + ⋯ |
| L(s) = 1 | − 1.00·2-s − 0.768·3-s + 0.00799·4-s − 1.55·5-s + 0.771·6-s − 1.60·7-s + 0.995·8-s − 0.409·9-s + 1.55·10-s + 0.301·11-s − 0.00614·12-s + 1.14·13-s + 1.61·14-s + 1.19·15-s − 1.00·16-s + 1.20·17-s + 0.410·18-s − 0.952·19-s − 0.0123·20-s + 1.23·21-s − 0.302·22-s − 0.765·24-s + 1.40·25-s − 1.14·26-s + 1.08·27-s − 0.0128·28-s − 0.820·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09646954077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09646954077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.41T + 2T^{2} \) |
| 3 | \( 1 + 1.33T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 4.25T + 7T^{2} \) |
| 13 | \( 1 - 4.11T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 + 5.73T + 31T^{2} \) |
| 37 | \( 1 - 8.00T + 37T^{2} \) |
| 41 | \( 1 + 8.40T + 41T^{2} \) |
| 43 | \( 1 + 2.20T + 43T^{2} \) |
| 47 | \( 1 - 5.39T + 47T^{2} \) |
| 53 | \( 1 - 1.58T + 53T^{2} \) |
| 59 | \( 1 - 2.29T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 + 11.8T + 67T^{2} \) |
| 71 | \( 1 + 2.81T + 71T^{2} \) |
| 73 | \( 1 - 8.31T + 73T^{2} \) |
| 79 | \( 1 + 7.13T + 79T^{2} \) |
| 83 | \( 1 - 1.35T + 83T^{2} \) |
| 89 | \( 1 + 2.64T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.181477056348562241670725300953, −7.51303354798690996172114692758, −6.80497888655191139400808649544, −6.13496385882756895699875159766, −5.39263512572136925626279341175, −4.21106455604718232997067417149, −3.73814424915109676119966523073, −2.98190444167717407720456336671, −1.25071486308553461942407333690, −0.22675069195514629928140932080,
0.22675069195514629928140932080, 1.25071486308553461942407333690, 2.98190444167717407720456336671, 3.73814424915109676119966523073, 4.21106455604718232997067417149, 5.39263512572136925626279341175, 6.13496385882756895699875159766, 6.80497888655191139400808649544, 7.51303354798690996172114692758, 8.181477056348562241670725300953
