| L(s) = 1 | − 2·7-s + 2·11-s − 13-s − 3·17-s + 2·19-s − 6·23-s + 29-s − 8·31-s − 37-s − 2·41-s − 10·43-s − 4·47-s − 3·49-s + 10·53-s + 4·59-s + 9·61-s + 14·67-s + 10·71-s + 9·73-s − 4·77-s + 10·79-s − 12·83-s + 11·89-s + 2·91-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.603·11-s − 0.277·13-s − 0.727·17-s + 0.458·19-s − 1.25·23-s + 0.185·29-s − 1.43·31-s − 0.164·37-s − 0.312·41-s − 1.52·43-s − 0.583·47-s − 3/7·49-s + 1.37·53-s + 0.520·59-s + 1.15·61-s + 1.71·67-s + 1.18·71-s + 1.05·73-s − 0.455·77-s + 1.12·79-s − 1.31·83-s + 1.16·89-s + 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.54542356614367, −13.89984477235497, −13.51018058354977, −12.93969616977407, −12.51198371228182, −11.97282793115552, −11.44688906806942, −11.06858578806835, −10.26634659781327, −9.838060863280513, −9.514883965516438, −8.848455525140596, −8.348767585557635, −7.811964786672133, −6.992737751257836, −6.732864844853123, −6.222670597474059, −5.436359086073970, −5.070625898676798, −4.187956893033775, −3.665166984068944, −3.268708569989944, −2.233580308995340, −1.905200908253443, −0.8078974052523228, 0,
0.8078974052523228, 1.905200908253443, 2.233580308995340, 3.268708569989944, 3.665166984068944, 4.187956893033775, 5.070625898676798, 5.436359086073970, 6.222670597474059, 6.732864844853123, 6.992737751257836, 7.811964786672133, 8.348767585557635, 8.848455525140596, 9.514883965516438, 9.838060863280513, 10.26634659781327, 11.06858578806835, 11.44688906806942, 11.97282793115552, 12.51198371228182, 12.93969616977407, 13.51018058354977, 13.89984477235497, 14.54542356614367
