| L(s) = 1 | + (0.405 + 2.19i)5-s − 3.23i·7-s − 1.54·11-s − 6.47i·13-s + 7.55i·17-s + 1.20·19-s + i·23-s + (−4.67 + 1.78i)25-s − 1.14·29-s − 6.97·31-s + (7.10 − 1.31i)35-s − 5.33i·37-s − 7.35·41-s − 3.63i·43-s − 10.3i·47-s + ⋯ |
| L(s) = 1 | + (0.181 + 0.983i)5-s − 1.22i·7-s − 0.464·11-s − 1.79i·13-s + 1.83i·17-s + 0.277·19-s + 0.208i·23-s + (−0.934 + 0.356i)25-s − 0.211·29-s − 1.25·31-s + (1.20 − 0.221i)35-s − 0.877i·37-s − 1.14·41-s − 0.554i·43-s − 1.50i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1942636590\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1942636590\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.405 - 2.19i)T \) |
| 23 | \( 1 - iT \) |
| good | 7 | \( 1 + 3.23iT - 7T^{2} \) |
| 11 | \( 1 + 1.54T + 11T^{2} \) |
| 13 | \( 1 + 6.47iT - 13T^{2} \) |
| 17 | \( 1 - 7.55iT - 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 29 | \( 1 + 1.14T + 29T^{2} \) |
| 31 | \( 1 + 6.97T + 31T^{2} \) |
| 37 | \( 1 + 5.33iT - 37T^{2} \) |
| 41 | \( 1 + 7.35T + 41T^{2} \) |
| 43 | \( 1 + 3.63iT - 43T^{2} \) |
| 47 | \( 1 + 10.3iT - 47T^{2} \) |
| 53 | \( 1 - 3.16iT - 53T^{2} \) |
| 59 | \( 1 + 8.15T + 59T^{2} \) |
| 61 | \( 1 - 0.160T + 61T^{2} \) |
| 67 | \( 1 - 15.1iT - 67T^{2} \) |
| 71 | \( 1 + 3.24T + 71T^{2} \) |
| 73 | \( 1 - 9.51iT - 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 - 3.26iT - 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 9.21iT - 97T^{2} \) |
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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
−7.904049003364759680203528944413, −7.38645536284637660464929951761, −6.74284729441168765066622902030, −5.76603236744414168855960955760, −5.33967926820519200830979682489, −3.87188710127314706725863691943, −3.61594372593802534436942129512, −2.58306346433145803124600808933, −1.44667244696725830634984769924, −0.05226301547639768587862429367,
1.52779791736151002113205937555, 2.30625722223189211123695811689, 3.26454042124610157914849916071, 4.58600037916135564557636902474, 4.92044003766219723342243504457, 5.71454321587756098142714416372, 6.50080297488980830238667419498, 7.33553038268707147529037586111, 8.150085270756566031936499038280, 8.968796920593463773742191862424
