L(s) = 1 | − 1.93i·3-s + (−2.32 − 10.9i)5-s + 7i·7-s + 23.2·9-s − 25.5·11-s − 64.1i·13-s + (−21.2 + 4.51i)15-s + 27.6i·17-s + 0.792·19-s + 13.5·21-s − 108. i·23-s + (−114. + 50.9i)25-s − 97.4i·27-s + 234.·29-s − 129.·31-s + ⋯ |
L(s) = 1 | − 0.373i·3-s + (−0.208 − 0.978i)5-s + 0.377i·7-s + 0.860·9-s − 0.700·11-s − 1.36i·13-s + (−0.365 + 0.0777i)15-s + 0.395i·17-s + 0.00956·19-s + 0.141·21-s − 0.984i·23-s + (−0.913 + 0.407i)25-s − 0.694i·27-s + 1.49·29-s − 0.748·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.978 + 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.053696768\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.053696768\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.32 + 10.9i)T \) |
| 7 | \( 1 - 7iT \) |
good | 3 | \( 1 + 1.93iT - 27T^{2} \) |
| 11 | \( 1 + 25.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.1iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 27.6iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 0.792T + 6.85e3T^{2} \) |
| 23 | \( 1 + 108. iT - 1.21e4T^{2} \) |
| 29 | \( 1 - 234.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 129.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 38.3iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 403.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 172. iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 206. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 144. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 679.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 574.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 515. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 556.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 173. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 79.3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.04e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 652.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 515. iT - 9.12e5T^{2} \) |
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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
−10.08178153434124443154038945489, −8.876405934548391451822673360952, −8.163002289574597322180443585377, −7.45228763522908269927513873875, −6.20734001431213556595844731131, −5.22335469470988763284171382873, −4.38294510986031494933877223390, −2.95517687442626099006353183729, −1.54355248770685609753885560826, −0.31320545496437749457231338719,
1.68927878622023975253544524171, 3.08580647142355475599630090587, 4.09575015570170317116377936919, 5.02038843069162909412396264012, 6.47603192646763346997200925931, 7.11222044747110897582088568461, 7.929597203833289142400525614325, 9.232482543492643403805239780460, 10.00774034313717126805884330753, 10.66603300570194355340319325960