L(s) = 1 | + (−1.34 − 0.442i)2-s + 0.901·3-s + (1.60 + 1.18i)4-s − i·5-s + (−1.21 − 0.398i)6-s + (−1.63 − 2.30i)8-s − 2.18·9-s + (−0.442 + 1.34i)10-s − 3.74i·11-s + (1.44 + 1.07i)12-s − 2.41i·13-s − 0.901i·15-s + (1.17 + 3.82i)16-s + 0.583i·17-s + (2.93 + 0.967i)18-s − 6.15·19-s + ⋯ |
L(s) = 1 | + (−0.949 − 0.312i)2-s + 0.520·3-s + (0.804 + 0.594i)4-s − 0.447i·5-s + (−0.494 − 0.162i)6-s + (−0.578 − 0.815i)8-s − 0.729·9-s + (−0.139 + 0.424i)10-s − 1.12i·11-s + (0.418 + 0.309i)12-s − 0.671i·13-s − 0.232i·15-s + (0.293 + 0.955i)16-s + 0.141i·17-s + (0.692 + 0.228i)18-s − 1.41·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.219i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0531970 - 0.479886i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0531970 - 0.479886i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.34 + 0.442i)T \) |
| 5 | \( 1 + iT \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 0.901T + 3T^{2} \) |
| 11 | \( 1 + 3.74iT - 11T^{2} \) |
| 13 | \( 1 + 2.41iT - 13T^{2} \) |
| 17 | \( 1 - 0.583iT - 17T^{2} \) |
| 19 | \( 1 + 6.15T + 19T^{2} \) |
| 23 | \( 1 - 4.31iT - 23T^{2} \) |
| 29 | \( 1 + 0.435T + 29T^{2} \) |
| 31 | \( 1 + 2.53T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 7.35iT - 41T^{2} \) |
| 43 | \( 1 - 5.80iT - 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 3.11T + 53T^{2} \) |
| 59 | \( 1 + 3.47T + 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 9.84iT - 67T^{2} \) |
| 71 | \( 1 + 9.96iT - 71T^{2} \) |
| 73 | \( 1 + 9.79iT - 73T^{2} \) |
| 79 | \( 1 - 0.459iT - 79T^{2} \) |
| 83 | \( 1 - 2.59T + 83T^{2} \) |
| 89 | \( 1 + 9.88iT - 89T^{2} \) |
| 97 | \( 1 + 4.54iT - 97T^{2} \) |
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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
−9.345004148675256275473297451884, −8.751115244845257722250537876485, −8.264156934544003534272977258245, −7.45749612585240397542413069397, −6.24592642514680063562564328713, −5.46757592071290150970073094772, −3.81084767406406126404374860421, −3.03169897197048999114015360029, −1.85583329910735896352097396050, −0.25970859540699021131754808282,
1.91972478006981381192670588866, 2.66734299043994615065085444124, 4.11699246518784113650723905306, 5.39747800383280773219734209300, 6.51211872700784935517209440819, 7.05575178018961337416718775092, 8.013441431276351695386297157808, 8.778836768592075447162140232341, 9.343742423631022788231540181766, 10.33928795614909088116921128683