L(s) = 1 | + 3-s + 9-s − 11-s − 6·13-s − 2·17-s − 4·19-s − 8·23-s + 27-s − 2·29-s − 33-s + 10·37-s − 6·39-s − 2·41-s − 8·43-s − 2·51-s + 10·53-s − 4·57-s − 4·59-s − 2·61-s − 12·67-s − 8·69-s + 4·71-s + 6·73-s − 12·79-s + 81-s + 12·83-s − 2·87-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s − 1.66·23-s + 0.192·27-s − 0.371·29-s − 0.174·33-s + 1.64·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s − 0.280·51-s + 1.37·53-s − 0.529·57-s − 0.520·59-s − 0.256·61-s − 1.46·67-s − 0.963·69-s + 0.474·71-s + 0.702·73-s − 1.35·79-s + 1/9·81-s + 1.31·83-s − 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−12.87133914492627, −12.42668937146021, −11.87347106662086, −11.71226805786700, −10.93687106329177, −10.45062160383449, −10.07428661852816, −9.666889603795428, −9.251701584293333, −8.689076615920053, −8.176447122712961, −7.813522657330041, −7.405107144798638, −6.837328939070340, −6.390819615367072, −5.800742741043458, −5.307372726753921, −4.562289316518812, −4.390589234592589, −3.808469891554616, −3.060935549939565, −2.598374724799270, −2.043010934050798, −1.762052956784514, −0.6128912901365485, 0,
0.6128912901365485, 1.762052956784514, 2.043010934050798, 2.598374724799270, 3.060935549939565, 3.808469891554616, 4.390589234592589, 4.562289316518812, 5.307372726753921, 5.800742741043458, 6.390819615367072, 6.837328939070340, 7.405107144798638, 7.813522657330041, 8.176447122712961, 8.689076615920053, 9.251701584293333, 9.666889603795428, 10.07428661852816, 10.45062160383449, 10.93687106329177, 11.71226805786700, 11.87347106662086, 12.42668937146021, 12.87133914492627