| L(s) = 1 | + 2.39·2-s + 3.73·4-s + 1.73·5-s + 1.75·7-s + 4.14·8-s + 4.14·10-s + 2·11-s + 2.46·13-s + 4.19·14-s + 2.46·16-s + 3.19·19-s + 6.46·20-s + 4.78·22-s − 1.53·23-s − 2.00·25-s + 5.89·26-s + 6.54·28-s − 1.73·29-s + 9.57·31-s − 2.39·32-s + 3.03·35-s − 10.0·37-s + 7.65·38-s + 7.18·40-s + 10.6·41-s − 3.73·43-s + 7.46·44-s + ⋯ |
| L(s) = 1 | + 1.69·2-s + 1.86·4-s + 0.774·5-s + 0.662·7-s + 1.46·8-s + 1.31·10-s + 0.603·11-s + 0.683·13-s + 1.12·14-s + 0.616·16-s + 0.733·19-s + 1.44·20-s + 1.02·22-s − 0.320·23-s − 0.400·25-s + 1.15·26-s + 1.23·28-s − 0.321·29-s + 1.72·31-s − 0.423·32-s + 0.513·35-s − 1.65·37-s + 1.24·38-s + 1.13·40-s + 1.66·41-s − 0.569·43-s + 1.12·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7803 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.293118226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.293118226\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - 2.39T + 2T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 2.46T + 13T^{2} \) |
| 19 | \( 1 - 3.19T + 19T^{2} \) |
| 23 | \( 1 + 1.53T + 23T^{2} \) |
| 29 | \( 1 + 1.73T + 29T^{2} \) |
| 31 | \( 1 - 9.57T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 3.73T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 3.03T + 59T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 - 7.73T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 3.03T + 73T^{2} \) |
| 79 | \( 1 - 7.82T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 13.0T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.68676312042546775389569948248, −6.70524150699525833152885679046, −6.34939632239923773364842618103, −5.54891211130720804574207155006, −5.16390021798114330520574733130, −4.27624149011911824441656667614, −3.73787013392988393287581521115, −2.85249728577045923299528853404, −2.03141536428772317914521004931, −1.24033268397176857425150351253,
1.24033268397176857425150351253, 2.03141536428772317914521004931, 2.85249728577045923299528853404, 3.73787013392988393287581521115, 4.27624149011911824441656667614, 5.16390021798114330520574733130, 5.54891211130720804574207155006, 6.34939632239923773364842618103, 6.70524150699525833152885679046, 7.68676312042546775389569948248
