L(s) = 1 | − 3-s − 4·7-s + 9-s − 11-s + 5·13-s + 17-s + 5·19-s + 4·21-s − 5·23-s − 27-s − 7·29-s + 9·31-s + 33-s + 8·37-s − 5·39-s − 10·41-s + 7·43-s − 8·47-s + 9·49-s − 51-s + 10·53-s − 5·57-s − 4·59-s − 2·61-s − 4·63-s − 10·67-s + 5·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.38·13-s + 0.242·17-s + 1.14·19-s + 0.872·21-s − 1.04·23-s − 0.192·27-s − 1.29·29-s + 1.61·31-s + 0.174·33-s + 1.31·37-s − 0.800·39-s − 1.56·41-s + 1.06·43-s − 1.16·47-s + 9/7·49-s − 0.140·51-s + 1.37·53-s − 0.662·57-s − 0.520·59-s − 0.256·61-s − 0.503·63-s − 1.22·67-s + 0.601·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 17 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.35704530659498, −12.70813066997404, −12.23656430581695, −11.69984631871461, −11.54902953534979, −10.70598096016778, −10.45165164778724, −9.922195239044431, −9.471439861777261, −9.199138087767866, −8.408452531823332, −7.991789894409610, −7.437562157510495, −6.919603269615648, −6.250235302672901, −6.118089353780988, −5.665687867788074, −5.030984078533402, −4.358553998756522, −3.755684632564327, −3.379425576847292, −2.874544478120515, −2.122387947837538, −1.299545240306477, −0.7358902946791664, 0,
0.7358902946791664, 1.299545240306477, 2.122387947837538, 2.874544478120515, 3.379425576847292, 3.755684632564327, 4.358553998756522, 5.030984078533402, 5.665687867788074, 6.118089353780988, 6.250235302672901, 6.919603269615648, 7.437562157510495, 7.991789894409610, 8.408452531823332, 9.199138087767866, 9.471439861777261, 9.922195239044431, 10.45165164778724, 10.70598096016778, 11.54902953534979, 11.69984631871461, 12.23656430581695, 12.70813066997404, 13.35704530659498