L(s) = 1 | − 2·3-s + 5-s + 7-s + 9-s − 2·11-s + 13-s − 2·15-s − 6·17-s + 6·19-s − 2·21-s + 6·23-s + 25-s + 4·27-s − 6·29-s − 10·31-s + 4·33-s + 35-s − 6·37-s − 2·39-s + 10·41-s + 2·43-s + 45-s + 49-s + 12·51-s − 10·53-s − 2·55-s − 12·57-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.603·11-s + 0.277·13-s − 0.516·15-s − 1.45·17-s + 1.37·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s − 1.79·31-s + 0.696·33-s + 0.169·35-s − 0.986·37-s − 0.320·39-s + 1.56·41-s + 0.304·43-s + 0.149·45-s + 1/7·49-s + 1.68·51-s − 1.37·53-s − 0.269·55-s − 1.58·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.091682339010621419829180547969, −7.24979488495703712107028344507, −6.65936778026373513818425588611, −5.70952700956783923844256466486, −5.31311820632342583936396912951, −4.64845669236501145277429513468, −3.49108300144622355409746697370, −2.39912736781171731142132851570, −1.29367808047020723985816078191, 0,
1.29367808047020723985816078191, 2.39912736781171731142132851570, 3.49108300144622355409746697370, 4.64845669236501145277429513468, 5.31311820632342583936396912951, 5.70952700956783923844256466486, 6.65936778026373513818425588611, 7.24979488495703712107028344507, 8.091682339010621419829180547969