| L(s) = 1 | + 3-s + 7-s + 9-s + 3.56·11-s − 5.68·13-s + 1.43·17-s + 7.12·19-s + 21-s + 4.12·23-s + 27-s − 0.123·29-s + 5.68·31-s + 3.56·33-s + 3.56·37-s − 5.68·39-s − 11.6·41-s − 8.12·43-s − 3.12·47-s + 49-s + 1.43·51-s − 4.56·53-s + 7.12·57-s + 9.68·59-s − 6.56·61-s + 63-s + 7.80·67-s + 4.12·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 0.333·9-s + 1.07·11-s − 1.57·13-s + 0.348·17-s + 1.63·19-s + 0.218·21-s + 0.859·23-s + 0.192·27-s − 0.0228·29-s + 1.02·31-s + 0.619·33-s + 0.585·37-s − 0.910·39-s − 1.82·41-s − 1.23·43-s − 0.455·47-s + 0.142·49-s + 0.201·51-s − 0.626·53-s + 0.943·57-s + 1.26·59-s − 0.840·61-s + 0.125·63-s + 0.953·67-s + 0.496·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.743559077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.743559077\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 3.56T + 11T^{2} \) |
| 13 | \( 1 + 5.68T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 - 4.12T + 23T^{2} \) |
| 29 | \( 1 + 0.123T + 29T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 - 3.56T + 37T^{2} \) |
| 41 | \( 1 + 11.6T + 41T^{2} \) |
| 43 | \( 1 + 8.12T + 43T^{2} \) |
| 47 | \( 1 + 3.12T + 47T^{2} \) |
| 53 | \( 1 + 4.56T + 53T^{2} \) |
| 59 | \( 1 - 9.68T + 59T^{2} \) |
| 61 | \( 1 + 6.56T + 61T^{2} \) |
| 67 | \( 1 - 7.80T + 67T^{2} \) |
| 71 | \( 1 + 7.80T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 8.43T + 79T^{2} \) |
| 83 | \( 1 - 9.68T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−8.342891979245086082225955556313, −7.69433885345837310586538888996, −7.04549668953965699247789720128, −6.40542899448805416256246386319, −5.08042057591114403961014270961, −4.87620393910601210683448169534, −3.64846973434820364175435893626, −3.01144146190462939960755721361, −1.97748224428306969702760413757, −0.960338179719121518290184192895,
0.960338179719121518290184192895, 1.97748224428306969702760413757, 3.01144146190462939960755721361, 3.64846973434820364175435893626, 4.87620393910601210683448169534, 5.08042057591114403961014270961, 6.40542899448805416256246386319, 7.04549668953965699247789720128, 7.69433885345837310586538888996, 8.342891979245086082225955556313
