L(s) = 1 | + 3-s − 7-s + 9-s + 2·11-s − 6·13-s + 2·19-s − 21-s − 23-s + 27-s − 6·29-s − 2·31-s + 2·33-s + 12·37-s − 6·39-s + 2·41-s + 8·43-s − 12·47-s + 49-s − 6·53-s + 2·57-s − 10·61-s − 63-s − 4·67-s − 69-s − 6·71-s − 2·73-s − 2·77-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.66·13-s + 0.458·19-s − 0.218·21-s − 0.208·23-s + 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.348·33-s + 1.97·37-s − 0.960·39-s + 0.312·41-s + 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.264·57-s − 1.28·61-s − 0.125·63-s − 0.488·67-s − 0.120·69-s − 0.712·71-s − 0.234·73-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.14251853882071, −13.58075808110809, −12.92437591993370, −12.71457386466391, −12.16862136340035, −11.54035380889254, −11.24040318244608, −10.46270204354956, −9.919727033161313, −9.475191637513621, −9.271079680362013, −8.662542036123472, −7.775974884039485, −7.587033668271800, −7.194947261524310, −6.324901243083288, −6.052125916477992, −5.249801782571133, −4.605580641198957, −4.266117791088401, −3.407928273818656, −3.034884657140726, −2.294086908456759, −1.806495903206662, −0.8790132887945035, 0,
0.8790132887945035, 1.806495903206662, 2.294086908456759, 3.034884657140726, 3.407928273818656, 4.266117791088401, 4.605580641198957, 5.249801782571133, 6.052125916477992, 6.324901243083288, 7.194947261524310, 7.587033668271800, 7.775974884039485, 8.662542036123472, 9.271079680362013, 9.475191637513621, 9.919727033161313, 10.46270204354956, 11.24040318244608, 11.54035380889254, 12.16862136340035, 12.71457386466391, 12.92437591993370, 13.58075808110809, 14.14251853882071