L(s) = 1 | − 1.41·2-s + (−0.158 − 1.72i)3-s + 2.00·4-s + (0.224 + 2.43i)6-s − 2.82·8-s + (−2.94 + 0.548i)9-s + 3.78i·11-s + (−0.317 − 3.44i)12-s + 4.00·16-s − 8.02·17-s + (4.17 − 0.775i)18-s − 6.34·19-s − 5.34i·22-s + (0.449 + 4.87i)24-s + (1.41 + 4.99i)27-s + ⋯ |
L(s) = 1 | − 1.00·2-s + (−0.0917 − 0.995i)3-s + 1.00·4-s + (0.0917 + 0.995i)6-s − 1.00·8-s + (−0.983 + 0.182i)9-s + 1.14i·11-s + (−0.0917 − 0.995i)12-s + 1.00·16-s − 1.94·17-s + (0.983 − 0.182i)18-s − 1.45·19-s − 1.14i·22-s + (0.0917 + 0.995i)24-s + (0.272 + 0.962i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0626621 + 0.112649i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0626621 + 0.112649i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41T \) |
| 3 | \( 1 + (0.158 + 1.72i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 3.78iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 8.02T + 17T^{2} \) |
| 19 | \( 1 + 6.34T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 10.8iT - 41T^{2} \) |
| 43 | \( 1 - 10iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 14.1iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 0.348iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 15.6iT - 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 + 18.4iT - 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−10.98635733053477978392522886992, −10.08705707634517204463396311927, −8.951077150044081440941298244130, −8.446176172849905530420777327093, −7.32514714363639507204022811662, −6.79214337252876949716857654502, −5.96153889278589183998733571304, −4.43068662218473563290185324129, −2.55974332666153819736857269023, −1.77178395313481482222082709951,
0.091392521812361061427253712997, 2.28980101745073913005824646575, 3.51173267978459936367321768806, 4.72583909243822732444860140078, 6.06255408423310718097842473570, 6.65530603530427155152855379083, 8.196149719819554742039198400008, 8.694513379201643017889231670840, 9.413402080148884617202143986953, 10.41670138438024737753513993496