| L(s) = 1 | + 19.8·7-s + 70.6·11-s + 0.761·13-s + 108.·17-s − 125.·19-s + 40.8·23-s + 140.·29-s + 296.·31-s + 26.8·37-s + 20.6·41-s − 32.5·43-s + 11.4·47-s + 52.2·49-s + 159.·53-s + 374.·59-s + 303.·61-s − 877.·67-s − 1.15e3·71-s − 120.·73-s + 1.40e3·77-s − 1.24e3·79-s + 654.·83-s − 795.·89-s + 15.1·91-s − 850.·97-s + 149.·101-s − 1.17e3·103-s + ⋯ |
| L(s) = 1 | + 1.07·7-s + 1.93·11-s + 0.0162·13-s + 1.54·17-s − 1.51·19-s + 0.370·23-s + 0.902·29-s + 1.72·31-s + 0.119·37-s + 0.0786·41-s − 0.115·43-s + 0.0354·47-s + 0.152·49-s + 0.413·53-s + 0.827·59-s + 0.636·61-s − 1.60·67-s − 1.93·71-s − 0.193·73-s + 2.07·77-s − 1.77·79-s + 0.865·83-s − 0.947·89-s + 0.0174·91-s − 0.890·97-s + 0.147·101-s − 1.12·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.309144327\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.309144327\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 19.8T + 343T^{2} \) |
| 11 | \( 1 - 70.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.761T + 2.19e3T^{2} \) |
| 17 | \( 1 - 108.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 40.8T + 1.21e4T^{2} \) |
| 29 | \( 1 - 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 296.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 26.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 20.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 32.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 11.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 159.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 374.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 303.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 877.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.15e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 120.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 654.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 795.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 850.T + 9.12e5T^{2} \) |
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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
−8.658648265088365022292382391090, −8.371556962720652617440891224682, −7.31853742830552253825849186505, −6.50650561418108627650056580061, −5.78530363004666736488681320534, −4.61764105235841312124844138776, −4.11780956899534261542317223276, −2.95271609250383857806876851806, −1.64183514423778633442674992939, −0.959590289230004109798269390635,
0.959590289230004109798269390635, 1.64183514423778633442674992939, 2.95271609250383857806876851806, 4.11780956899534261542317223276, 4.61764105235841312124844138776, 5.78530363004666736488681320534, 6.50650561418108627650056580061, 7.31853742830552253825849186505, 8.371556962720652617440891224682, 8.658648265088365022292382391090
