L(s) = 1 | + 3-s + 9-s + 4·11-s + 6·13-s + 6·17-s − 4·19-s + 4·23-s + 27-s + 2·29-s − 8·31-s + 4·33-s + 6·37-s + 6·39-s − 6·41-s + 8·43-s + 6·51-s + 6·53-s − 4·57-s − 4·59-s + 10·61-s + 8·67-s + 4·69-s + 12·71-s − 14·73-s − 16·79-s + 81-s + 12·83-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 1.45·17-s − 0.917·19-s + 0.834·23-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.696·33-s + 0.986·37-s + 0.960·39-s − 0.937·41-s + 1.21·43-s + 0.840·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s + 1.28·61-s + 0.977·67-s + 0.481·69-s + 1.42·71-s − 1.63·73-s − 1.80·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.722649873\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.722649873\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.90261073217729, −12.63495247268480, −11.93725278405716, −11.50365286996099, −11.07054967413526, −10.61558654614314, −10.08125954595713, −9.580184005318574, −9.041516067753061, −8.734667590608395, −8.334298218926753, −7.786246631280687, −7.174815597747929, −6.807950953891014, −6.152203490503090, −5.844001602809484, −5.249806391745321, −4.485795226793745, −3.948236001525937, −3.605052855024702, −3.176677077354809, −2.406843801675392, −1.712210069255680, −1.177439714077975, −0.7106885259938791,
0.7106885259938791, 1.177439714077975, 1.712210069255680, 2.406843801675392, 3.176677077354809, 3.605052855024702, 3.948236001525937, 4.485795226793745, 5.249806391745321, 5.844001602809484, 6.152203490503090, 6.807950953891014, 7.174815597747929, 7.786246631280687, 8.334298218926753, 8.734667590608395, 9.041516067753061, 9.580184005318574, 10.08125954595713, 10.61558654614314, 11.07054967413526, 11.50365286996099, 11.93725278405716, 12.63495247268480, 12.90261073217729