| L(s) = 1 | + 3-s − 2·9-s − 11-s − 6·13-s + 7·17-s − 19-s + 8·23-s − 5·27-s − 6·29-s − 4·31-s − 33-s + 8·37-s − 6·39-s + 5·41-s − 6·47-s + 7·51-s + 4·53-s − 57-s + 4·59-s − 6·61-s − 5·67-s + 8·69-s + 14·71-s − 15·73-s + 14·79-s + 81-s − 83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 2/3·9-s − 0.301·11-s − 1.66·13-s + 1.69·17-s − 0.229·19-s + 1.66·23-s − 0.962·27-s − 1.11·29-s − 0.718·31-s − 0.174·33-s + 1.31·37-s − 0.960·39-s + 0.780·41-s − 0.875·47-s + 0.980·51-s + 0.549·53-s − 0.132·57-s + 0.520·59-s − 0.768·61-s − 0.610·67-s + 0.963·69-s + 1.66·71-s − 1.75·73-s + 1.57·79-s + 1/9·81-s − 0.109·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.954397015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.954397015\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−7.56494180922572187938335857187, −7.35500787365478227999287072808, −6.29325670024540677134081710625, −5.40767559028498159400668896490, −5.13571387993301717631254731855, −4.11005975680330660518619305488, −3.17551447290942259041284734935, −2.76502253204695250874045311669, −1.89027430603880483807337724759, −0.62360942617365088824194803159,
0.62360942617365088824194803159, 1.89027430603880483807337724759, 2.76502253204695250874045311669, 3.17551447290942259041284734935, 4.11005975680330660518619305488, 5.13571387993301717631254731855, 5.40767559028498159400668896490, 6.29325670024540677134081710625, 7.35500787365478227999287072808, 7.56494180922572187938335857187
