| L(s) = 1 | − 34·7-s − 18·11-s − 12·13-s − 106·17-s − 44·19-s + 56·23-s − 270·29-s + 204·31-s − 120·37-s − 80·41-s − 536·43-s − 536·47-s + 813·49-s + 542·53-s + 174·59-s + 186·61-s − 332·67-s + 132·71-s + 602·73-s + 612·77-s − 548·79-s − 492·83-s + 1.05e3·89-s + 408·91-s − 482·97-s − 1.21e3·101-s − 898·103-s + ⋯ |
| L(s) = 1 | − 1.83·7-s − 0.493·11-s − 0.256·13-s − 1.51·17-s − 0.531·19-s + 0.507·23-s − 1.72·29-s + 1.18·31-s − 0.533·37-s − 0.304·41-s − 1.90·43-s − 1.66·47-s + 2.37·49-s + 1.40·53-s + 0.383·59-s + 0.390·61-s − 0.605·67-s + 0.220·71-s + 0.965·73-s + 0.905·77-s − 0.780·79-s − 0.650·83-s + 1.25·89-s + 0.470·91-s − 0.504·97-s − 1.19·101-s − 0.859·103-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
| \(L(2)\) |
\(\approx\) |
\(0.4968437134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4968437134\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 34 T + p^{3} T^{2} \) |
| 11 | \( 1 + 18 T + p^{3} T^{2} \) |
| 13 | \( 1 + 12 T + p^{3} T^{2} \) |
| 17 | \( 1 + 106 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 + 270 T + p^{3} T^{2} \) |
| 31 | \( 1 - 204 T + p^{3} T^{2} \) |
| 37 | \( 1 + 120 T + p^{3} T^{2} \) |
| 41 | \( 1 + 80 T + p^{3} T^{2} \) |
| 43 | \( 1 + 536 T + p^{3} T^{2} \) |
| 47 | \( 1 + 536 T + p^{3} T^{2} \) |
| 53 | \( 1 - 542 T + p^{3} T^{2} \) |
| 59 | \( 1 - 174 T + p^{3} T^{2} \) |
| 61 | \( 1 - 186 T + p^{3} T^{2} \) |
| 67 | \( 1 + 332 T + p^{3} T^{2} \) |
| 71 | \( 1 - 132 T + p^{3} T^{2} \) |
| 73 | \( 1 - 602 T + p^{3} T^{2} \) |
| 79 | \( 1 + 548 T + p^{3} T^{2} \) |
| 83 | \( 1 + 492 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1052 T + p^{3} T^{2} \) |
| 97 | \( 1 + 482 T + p^{3} T^{2} \) |
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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
−8.977314511360815655529520994354, −8.265353463563269687997126423904, −6.99554307713068392964605740925, −6.70700217952046089972331297420, −5.81496300546849744815301204774, −4.82061752854122721702733716237, −3.77749210464564741815954163001, −2.97919958800080101311492941657, −2.04650628620539726469342120586, −0.30814857192914688972266718032,
0.30814857192914688972266718032, 2.04650628620539726469342120586, 2.97919958800080101311492941657, 3.77749210464564741815954163001, 4.82061752854122721702733716237, 5.81496300546849744815301204774, 6.70700217952046089972331297420, 6.99554307713068392964605740925, 8.265353463563269687997126423904, 8.977314511360815655529520994354
