| L(s) = 1 | − 7-s − 2·17-s + 4·19-s + 4·23-s − 5·25-s + 6·29-s + 4·31-s + 6·37-s − 6·41-s + 4·43-s + 12·47-s + 49-s − 4·53-s − 8·61-s + 12·67-s − 4·71-s + 4·73-s + 4·83-s − 12·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 2·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.485·17-s + 0.917·19-s + 0.834·23-s − 25-s + 1.11·29-s + 0.718·31-s + 0.986·37-s − 0.937·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.549·53-s − 1.02·61-s + 1.46·67-s − 0.474·71-s + 0.468·73-s + 0.439·83-s − 1.27·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.309865176\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.309865176\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.17375222775317, −13.70384600799176, −13.44814610133441, −12.77630146853069, −12.14950116021803, −11.94788254727551, −11.13974419860957, −10.85472928429197, −10.13605486275189, −9.636763213174008, −9.289250232283687, −8.599747788536208, −8.088159618798430, −7.537099612699949, −6.894418257131551, −6.514183774345589, −5.769286563814259, −5.375084720649882, −4.525158343104896, −4.186691853362645, −3.297594512027262, −2.835937952515950, −2.166261229285870, −1.227823113904556, −0.5616856939372839,
0.5616856939372839, 1.227823113904556, 2.166261229285870, 2.835937952515950, 3.297594512027262, 4.186691853362645, 4.525158343104896, 5.375084720649882, 5.769286563814259, 6.514183774345589, 6.894418257131551, 7.537099612699949, 8.088159618798430, 8.599747788536208, 9.289250232283687, 9.636763213174008, 10.13605486275189, 10.85472928429197, 11.13974419860957, 11.94788254727551, 12.14950116021803, 12.77630146853069, 13.44814610133441, 13.70384600799176, 14.17375222775317
