L(s) = 1 | + (2.17 + 0.523i)5-s − 4.50i·7-s + 4.10·11-s − 4.10i·13-s − 2.26i·17-s − 6.77·19-s − i·23-s + (4.45 + 2.27i)25-s + 4.13·29-s + 1.84·31-s + (2.36 − 9.80i)35-s − 11.1i·37-s − 8.36·41-s + 5.43i·43-s − 0.593i·47-s + ⋯ |
L(s) = 1 | + (0.972 + 0.234i)5-s − 1.70i·7-s + 1.23·11-s − 1.13i·13-s − 0.549i·17-s − 1.55·19-s − 0.208i·23-s + (0.890 + 0.455i)25-s + 0.768·29-s + 0.330·31-s + (0.398 − 1.65i)35-s − 1.82i·37-s − 1.30·41-s + 0.828i·43-s − 0.0865i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.179438662\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.179438662\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.17 - 0.523i)T \) |
| 23 | \( 1 + iT \) |
good | 7 | \( 1 + 4.50iT - 7T^{2} \) |
| 11 | \( 1 - 4.10T + 11T^{2} \) |
| 13 | \( 1 + 4.10iT - 13T^{2} \) |
| 17 | \( 1 + 2.26iT - 17T^{2} \) |
| 19 | \( 1 + 6.77T + 19T^{2} \) |
| 29 | \( 1 - 4.13T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + 11.1iT - 37T^{2} \) |
| 41 | \( 1 + 8.36T + 41T^{2} \) |
| 43 | \( 1 - 5.43iT - 43T^{2} \) |
| 47 | \( 1 + 0.593iT - 47T^{2} \) |
| 53 | \( 1 + 1.70iT - 53T^{2} \) |
| 59 | \( 1 - 6.19T + 59T^{2} \) |
| 61 | \( 1 + 11.3T + 61T^{2} \) |
| 67 | \( 1 - 5.78iT - 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 0.363iT - 73T^{2} \) |
| 79 | \( 1 + 1.75T + 79T^{2} \) |
| 83 | \( 1 - 9.72iT - 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 + 4.38iT - 97T^{2} \) |
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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
−8.217537146787565725727420762699, −7.31600539424666295431267189991, −6.69260709873358566540238328339, −6.22815950395190750672705019168, −5.20634842908520667994851773231, −4.33222734227682177350682848716, −3.67671678333987944508475167212, −2.66952635290840115429642714801, −1.52938309788626654142120747812, −0.60066530396457327524693845423,
1.57064466168058140356831009471, 2.04507785988943521520266937616, 3.02349260295233021903107286889, 4.22497717799505504769229366749, 4.92815686196758609087193877345, 5.81195636712125609113208365294, 6.52323653015708853048293426234, 6.65107634145747160750100303193, 8.398907683361077012022444175663, 8.584230725264262250377902459105