L(s) = 1 | − 1.04i·2-s − 0.531i·3-s + 0.902·4-s + (1.65 − 1.50i)5-s − 0.556·6-s − 2.74i·7-s − 3.04i·8-s + 2.71·9-s + (−1.57 − 1.73i)10-s + 0.832·11-s − 0.479i·12-s − 0.610i·13-s − 2.87·14-s + (−0.800 − 0.877i)15-s − 1.37·16-s + 4.83i·17-s + ⋯ |
L(s) = 1 | − 0.740i·2-s − 0.306i·3-s + 0.451·4-s + (0.738 − 0.673i)5-s − 0.227·6-s − 1.03i·7-s − 1.07i·8-s + 0.905·9-s + (−0.499 − 0.547i)10-s + 0.251·11-s − 0.138i·12-s − 0.169i·13-s − 0.767·14-s + (−0.206 − 0.226i)15-s − 0.344·16-s + 1.17i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.621796978\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.621796978\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-1.65 + 1.50i)T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 1.04iT - 2T^{2} \) |
| 3 | \( 1 + 0.531iT - 3T^{2} \) |
| 7 | \( 1 + 2.74iT - 7T^{2} \) |
| 11 | \( 1 - 0.832T + 11T^{2} \) |
| 13 | \( 1 + 0.610iT - 13T^{2} \) |
| 17 | \( 1 - 4.83iT - 17T^{2} \) |
| 23 | \( 1 + 3.75iT - 23T^{2} \) |
| 29 | \( 1 - 3.97T + 29T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 - 4.33iT - 37T^{2} \) |
| 41 | \( 1 - 5.31T + 41T^{2} \) |
| 43 | \( 1 - 10.4iT - 43T^{2} \) |
| 47 | \( 1 - 3.40iT - 47T^{2} \) |
| 53 | \( 1 - 13.6iT - 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 9.07T + 61T^{2} \) |
| 67 | \( 1 - 10.9iT - 67T^{2} \) |
| 71 | \( 1 + 0.677T + 71T^{2} \) |
| 73 | \( 1 + 7.01iT - 73T^{2} \) |
| 79 | \( 1 + 3.47T + 79T^{2} \) |
| 83 | \( 1 - 4.97iT - 83T^{2} \) |
| 89 | \( 1 - 5.88T + 89T^{2} \) |
| 97 | \( 1 + 12.4iT - 97T^{2} \) |
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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
−9.225234320820718063485656878994, −8.143727453942252105989494673514, −7.35173224551082936657579776195, −6.55901090545799221635771316340, −5.94474759670010324891290690539, −4.51465559558716460050515107411, −4.00703716007821906348018997510, −2.73351192642314388735976543719, −1.57731315511407829231185876902, −1.04478485693328233626836055214,
1.81043276448645367723951790668, 2.55448244410996841148406212285, 3.66745385958929722713050697489, 5.14382204489898012483845864506, 5.52428568759313228466654693757, 6.51952618699298928813546875080, 7.04949026532880443317325580171, 7.74116791775195077704265855020, 8.950758007797051526505075302740, 9.391834011887580513848135774649