| L(s) = 1 | + 3-s − 3·7-s + 9-s + 2·13-s − 2·17-s − 2·19-s − 3·21-s − 7·23-s + 27-s + 2·29-s + 7·31-s − 2·37-s + 2·39-s + 3·41-s + 4·43-s + 47-s + 2·49-s − 2·51-s − 10·53-s − 2·57-s − 4·59-s − 3·63-s − 7·69-s − 13·73-s + 12·79-s + 81-s + 2·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.554·13-s − 0.485·17-s − 0.458·19-s − 0.654·21-s − 1.45·23-s + 0.192·27-s + 0.371·29-s + 1.25·31-s − 0.328·37-s + 0.320·39-s + 0.468·41-s + 0.609·43-s + 0.145·47-s + 2/7·49-s − 0.280·51-s − 1.37·53-s − 0.264·57-s − 0.520·59-s − 0.377·63-s − 0.842·69-s − 1.52·73-s + 1.35·79-s + 1/9·81-s + 0.219·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−13.66107111073730, −13.13796527837272, −12.70917691704617, −12.31623862612254, −11.79145726409908, −11.19738513789622, −10.61169978959330, −10.20636166691729, −9.691182696965369, −9.378433873816296, −8.636816305762949, −8.459976669891763, −7.760574479955253, −7.329783097965771, −6.607549635171234, −6.147878244501446, −6.039849030274104, −5.037465963472930, −4.397603495432189, −4.021946018892564, −3.343082239545589, −2.929288651857910, −2.270758159076263, −1.687075547116732, −0.7882517243257391, 0,
0.7882517243257391, 1.687075547116732, 2.270758159076263, 2.929288651857910, 3.343082239545589, 4.021946018892564, 4.397603495432189, 5.037465963472930, 6.039849030274104, 6.147878244501446, 6.607549635171234, 7.329783097965771, 7.760574479955253, 8.459976669891763, 8.636816305762949, 9.378433873816296, 9.691182696965369, 10.20636166691729, 10.61169978959330, 11.19738513789622, 11.79145726409908, 12.31623862612254, 12.70917691704617, 13.13796527837272, 13.66107111073730
