| L(s) = 1 | − 0.656·2-s − 1.56·4-s − 3.56·5-s + 2.34·8-s + 2.34·10-s + 11-s + 5.91·13-s + 1.59·16-s − 1.65·17-s − 1.48·19-s + 5.59·20-s − 0.656·22-s − 3.34·23-s + 7.73·25-s − 3.88·26-s − 3.08·29-s + 7.08·31-s − 5.73·32-s + 1.08·34-s − 4.51·37-s + 0.972·38-s − 8.36·40-s − 1.28·41-s + 1.59·43-s − 1.56·44-s + 2.19·46-s − 1.65·47-s + ⋯ |
| L(s) = 1 | − 0.464·2-s − 0.784·4-s − 1.59·5-s + 0.828·8-s + 0.741·10-s + 0.301·11-s + 1.63·13-s + 0.399·16-s − 0.401·17-s − 0.339·19-s + 1.25·20-s − 0.139·22-s − 0.697·23-s + 1.54·25-s − 0.761·26-s − 0.571·29-s + 1.27·31-s − 1.01·32-s + 0.186·34-s − 0.741·37-s + 0.157·38-s − 1.32·40-s − 0.200·41-s + 0.243·43-s − 0.236·44-s + 0.323·46-s − 0.241·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6746135238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6746135238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.656T + 2T^{2} \) |
| 5 | \( 1 + 3.56T + 5T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 17 | \( 1 + 1.65T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 + 3.34T + 23T^{2} \) |
| 29 | \( 1 + 3.08T + 29T^{2} \) |
| 31 | \( 1 - 7.08T + 31T^{2} \) |
| 37 | \( 1 + 4.51T + 37T^{2} \) |
| 41 | \( 1 + 1.28T + 41T^{2} \) |
| 43 | \( 1 - 1.59T + 43T^{2} \) |
| 47 | \( 1 + 1.65T + 47T^{2} \) |
| 53 | \( 1 + 9.22T + 53T^{2} \) |
| 59 | \( 1 - 8.85T + 59T^{2} \) |
| 61 | \( 1 - 6.68T + 61T^{2} \) |
| 67 | \( 1 + 9.82T + 67T^{2} \) |
| 71 | \( 1 - 8.61T + 71T^{2} \) |
| 73 | \( 1 + 4.56T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 - 0.167T + 83T^{2} \) |
| 89 | \( 1 + 2.56T + 89T^{2} \) |
| 97 | \( 1 + 9.73T + 97T^{2} \) |
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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
−8.344011682363531776874840457652, −7.85744301124524372313449093489, −6.98517184990413893678516334225, −6.22887840491009776233946266571, −5.21438128436112609285710543649, −4.18842889087189281204322866383, −4.01349874302529494242384755989, −3.15353508231980236838488783745, −1.55435385167247344933819242208, −0.51694091230656924347063307889,
0.51694091230656924347063307889, 1.55435385167247344933819242208, 3.15353508231980236838488783745, 4.01349874302529494242384755989, 4.18842889087189281204322866383, 5.21438128436112609285710543649, 6.22887840491009776233946266571, 6.98517184990413893678516334225, 7.85744301124524372313449093489, 8.344011682363531776874840457652
