L(s) = 1 | + (1.37 + 0.315i)2-s + (1.80 + 0.869i)4-s + (−2.03 − 0.937i)5-s + 3.36i·7-s + (2.20 + 1.76i)8-s + (−2.50 − 1.93i)10-s + 4.10i·11-s − 6.13·13-s + (−1.06 + 4.64i)14-s + (2.48 + 3.13i)16-s − 7.47i·17-s + 6.20i·19-s + (−2.84 − 3.45i)20-s + (−1.29 + 5.65i)22-s + 3.15i·23-s + ⋯ |
L(s) = 1 | + (0.974 + 0.223i)2-s + (0.900 + 0.434i)4-s + (−0.907 − 0.419i)5-s + 1.27i·7-s + (0.780 + 0.624i)8-s + (−0.791 − 0.611i)10-s + 1.23i·11-s − 1.70·13-s + (−0.284 + 1.24i)14-s + (0.621 + 0.783i)16-s − 1.81i·17-s + 1.42i·19-s + (−0.635 − 0.772i)20-s + (−0.275 + 1.20i)22-s + 0.658i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.002159406\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.002159406\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.37 - 0.315i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.03 + 0.937i)T \) |
good | 7 | \( 1 - 3.36iT - 7T^{2} \) |
| 11 | \( 1 - 4.10iT - 11T^{2} \) |
| 13 | \( 1 + 6.13T + 13T^{2} \) |
| 17 | \( 1 + 7.47iT - 17T^{2} \) |
| 19 | \( 1 - 6.20iT - 19T^{2} \) |
| 23 | \( 1 - 3.15iT - 23T^{2} \) |
| 29 | \( 1 - 1.34iT - 29T^{2} \) |
| 31 | \( 1 - 0.229T + 31T^{2} \) |
| 37 | \( 1 + 5.64T + 37T^{2} \) |
| 41 | \( 1 - 4.52T + 41T^{2} \) |
| 43 | \( 1 + 2.02T + 43T^{2} \) |
| 47 | \( 1 - 1.67iT - 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 3.07iT - 59T^{2} \) |
| 61 | \( 1 + 0.762iT - 61T^{2} \) |
| 67 | \( 1 - 13.8T + 67T^{2} \) |
| 71 | \( 1 - 1.26T + 71T^{2} \) |
| 73 | \( 1 - 13.9iT - 73T^{2} \) |
| 79 | \( 1 - 0.245T + 79T^{2} \) |
| 83 | \( 1 + 13.3T + 83T^{2} \) |
| 89 | \( 1 - 8.64T + 89T^{2} \) |
| 97 | \( 1 + 12.2iT - 97T^{2} \) |
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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
−10.05406903734641898276796567411, −9.356642150689163691309554361204, −8.264448839136443428535194306839, −7.39599832104879343650550945280, −6.99417190823488475174615241787, −5.45731225861278412283255706116, −5.08711396753731259044010233722, −4.18734220871484367226425398568, −2.95388268790790232465424389010, −2.05673562894866175843236899097,
0.61422729467338327455922447667, 2.46207474425366009135045992942, 3.51543469454665196000717815522, 4.19038912500212753519179849296, 5.03904972398373144017992938861, 6.29590652188338836337164116502, 7.03128669567242154295977323862, 7.65343339873855965870468588529, 8.616555893253297686847576882003, 10.07744974860700445203958344465