L(s) = 1 | + 2-s + (−1.68 − 0.420i)3-s + 4-s + (1.08 + 1.95i)5-s + (−1.68 − 0.420i)6-s + (0.595 + 2.57i)7-s + 8-s + (2.64 + 1.41i)9-s + (1.08 + 1.95i)10-s − 2.82i·11-s + (−1.68 − 0.420i)12-s + 3.36·13-s + (0.595 + 2.57i)14-s + (−1 − 3.74i)15-s + 16-s − 4.75i·17-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.970 − 0.242i)3-s + 0.5·4-s + (0.485 + 0.874i)5-s + (−0.685 − 0.171i)6-s + (0.224 + 0.974i)7-s + 0.353·8-s + (0.881 + 0.471i)9-s + (0.343 + 0.618i)10-s − 0.852i·11-s + (−0.485 − 0.121i)12-s + 0.931·13-s + (0.159 + 0.688i)14-s + (−0.258 − 0.966i)15-s + 0.250·16-s − 1.15i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45970 + 0.363453i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45970 + 0.363453i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + (1.68 + 0.420i)T \) |
| 5 | \( 1 + (-1.08 - 1.95i)T \) |
| 7 | \( 1 + (-0.595 - 2.57i)T \) |
good | 11 | \( 1 + 2.82iT - 11T^{2} \) |
| 13 | \( 1 - 3.36T + 13T^{2} \) |
| 17 | \( 1 + 4.75iT - 17T^{2} \) |
| 19 | \( 1 - 5.59iT - 19T^{2} \) |
| 23 | \( 1 + 7.29T + 23T^{2} \) |
| 29 | \( 1 - 0.500iT - 29T^{2} \) |
| 31 | \( 1 + 3.06iT - 31T^{2} \) |
| 37 | \( 1 + 3.32iT - 37T^{2} \) |
| 41 | \( 1 + 4.33T + 41T^{2} \) |
| 43 | \( 1 + 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 7.82iT - 47T^{2} \) |
| 53 | \( 1 - 8.58T + 53T^{2} \) |
| 59 | \( 1 - 2.16T + 59T^{2} \) |
| 61 | \( 1 + 2.52iT - 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 + 9.81iT - 71T^{2} \) |
| 73 | \( 1 - 5.53T + 73T^{2} \) |
| 79 | \( 1 - 3.29T + 79T^{2} \) |
| 83 | \( 1 - 6.97iT - 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−12.21569496258664607393584433417, −11.63323282010065916909796753037, −10.81131762723626001337174884613, −9.857982235892728360473628902166, −8.276463861135320021997353473805, −6.96254946178784730539025155305, −5.88099972654355601767906937151, −5.55109237085821038101068658167, −3.72393751990918871139912241278, −2.09773636479614506714933654805,
1.46644578542261602795297905447, 4.05085180131574216898327624018, 4.73279922876582492350683131456, 5.91866758129159635228733639978, 6.81519823234917862555710953708, 8.183019380567621115488490942066, 9.690257343409448732607526137265, 10.50621479055246978482605209831, 11.40469293514864609771462746373, 12.43337890134681571774601774094