| L(s) = 1 | + 2.23·3-s + 4.47·7-s + 2.00·9-s − 2.23·11-s − 4·13-s + 7·17-s + 6.70·19-s + 10.0·21-s − 4.47·23-s − 2.23·27-s − 4.47·31-s − 5.00·33-s − 2·37-s − 8.94·39-s + 5·41-s − 8.94·47-s + 13.0·49-s + 15.6·51-s − 6·53-s + 15.0·57-s − 8.94·59-s + 10·61-s + 8.94·63-s − 2.23·67-s − 10.0·69-s + 8.94·71-s + 9·73-s + ⋯ |
| L(s) = 1 | + 1.29·3-s + 1.69·7-s + 0.666·9-s − 0.674·11-s − 1.10·13-s + 1.69·17-s + 1.53·19-s + 2.18·21-s − 0.932·23-s − 0.430·27-s − 0.803·31-s − 0.870·33-s − 0.328·37-s − 1.43·39-s + 0.780·41-s − 1.30·47-s + 1.85·49-s + 2.19·51-s − 0.824·53-s + 1.98·57-s − 1.16·59-s + 1.28·61-s + 1.12·63-s − 0.273·67-s − 1.20·69-s + 1.06·71-s + 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.662465387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.662465387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.23T + 3T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 - 7T + 17T^{2} \) |
| 19 | \( 1 - 6.70T + 19T^{2} \) |
| 23 | \( 1 + 4.47T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 8.94T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 2.23T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 - 9T + 73T^{2} \) |
| 79 | \( 1 - 4.47T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 5T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−9.986432052311729962297171042333, −9.469037903208804314544292847471, −8.293278471731449323748831653016, −7.79561581782530048929272172623, −7.42178900017990312313869102370, −5.52904364062413850940138268946, −4.94212083798423090161146530053, −3.64077437817873934130839927969, −2.62290240944353820464194375718, −1.56449069602260475061450308551,
1.56449069602260475061450308551, 2.62290240944353820464194375718, 3.64077437817873934130839927969, 4.94212083798423090161146530053, 5.52904364062413850940138268946, 7.42178900017990312313869102370, 7.79561581782530048929272172623, 8.293278471731449323748831653016, 9.469037903208804314544292847471, 9.986432052311729962297171042333
