L(s) = 1 | + 4·7-s − 4·11-s − 6·13-s + 2·17-s + 4·19-s − 10·29-s − 4·31-s + 10·37-s − 2·41-s + 4·43-s + 8·47-s + 9·49-s + 2·53-s − 12·59-s − 10·61-s − 12·67-s − 10·73-s − 16·77-s − 4·79-s + 4·83-s + 6·89-s − 24·91-s + 14·97-s − 2·101-s − 4·103-s − 12·107-s − 2·109-s + ⋯ |
L(s) = 1 | + 1.51·7-s − 1.20·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 1.85·29-s − 0.718·31-s + 1.64·37-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s + 0.274·53-s − 1.56·59-s − 1.28·61-s − 1.46·67-s − 1.17·73-s − 1.82·77-s − 0.450·79-s + 0.439·83-s + 0.635·89-s − 2.51·91-s + 1.42·97-s − 0.199·101-s − 0.394·103-s − 1.16·107-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−7.63549292897984575899131203626, −7.35789374056298622274879760712, −5.91786855315071614709856382668, −5.34406571842942474718305872011, −4.86603791683645256704437945563, −4.14349220208172372443411887835, −2.95146574265753798599806944651, −2.28168724452191259317974952476, −1.38728363501538575831726414948, 0,
1.38728363501538575831726414948, 2.28168724452191259317974952476, 2.95146574265753798599806944651, 4.14349220208172372443411887835, 4.86603791683645256704437945563, 5.34406571842942474718305872011, 5.91786855315071614709856382668, 7.35789374056298622274879760712, 7.63549292897984575899131203626