| L(s) = 1 | − 4·7-s − 11-s + 17-s − 7·19-s + 6·23-s − 2·29-s + 10·31-s − 8·37-s + 5·41-s + 5·43-s − 4·47-s + 9·49-s + 10·53-s + 9·59-s − 10·61-s + 3·67-s − 6·71-s + 11·73-s + 4·77-s + 10·79-s + 4·83-s − 18·89-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.301·11-s + 0.242·17-s − 1.60·19-s + 1.25·23-s − 0.371·29-s + 1.79·31-s − 1.31·37-s + 0.780·41-s + 0.762·43-s − 0.583·47-s + 9/7·49-s + 1.37·53-s + 1.17·59-s − 1.28·61-s + 0.366·67-s − 0.712·71-s + 1.28·73-s + 0.455·77-s + 1.12·79-s + 0.439·83-s − 1.90·89-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.188975047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.188975047\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−14.10737267574075, −13.64420977256419, −13.16985457612533, −12.66822897783568, −12.44815768013055, −11.79691686508104, −11.07185755642010, −10.63694722993403, −10.11659993374844, −9.748919756132755, −9.027674778125814, −8.722091720483545, −8.093731143522155, −7.381374802578964, −6.805366254380570, −6.454214127161482, −5.928521655024046, −5.258937763386394, −4.613064197984708, −3.929085784768980, −3.417807700168452, −2.680897804435617, −2.330261122592645, −1.193220861211701, −0.3914088102414002,
0.3914088102414002, 1.193220861211701, 2.330261122592645, 2.680897804435617, 3.417807700168452, 3.929085784768980, 4.613064197984708, 5.258937763386394, 5.928521655024046, 6.454214127161482, 6.805366254380570, 7.381374802578964, 8.093731143522155, 8.722091720483545, 9.027674778125814, 9.748919756132755, 10.11659993374844, 10.63694722993403, 11.07185755642010, 11.79691686508104, 12.44815768013055, 12.66822897783568, 13.16985457612533, 13.64420977256419, 14.10737267574075
