| L(s) = 1 | − 2·3-s − 5-s − 4·7-s + 9-s − 2·13-s + 2·15-s + 4·17-s + 8·21-s + 25-s + 4·27-s + 4·35-s + 4·37-s + 4·39-s + 2·41-s + 2·43-s − 45-s + 10·47-s + 9·49-s − 8·51-s − 6·53-s + 4·59-s + 6·61-s − 4·63-s + 2·65-s + 8·71-s − 8·73-s − 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s + 0.970·17-s + 1.74·21-s + 1/5·25-s + 0.769·27-s + 0.676·35-s + 0.657·37-s + 0.640·39-s + 0.312·41-s + 0.304·43-s − 0.149·45-s + 1.45·47-s + 9/7·49-s − 1.12·51-s − 0.824·53-s + 0.520·59-s + 0.768·61-s − 0.503·63-s + 0.248·65-s + 0.949·71-s − 0.936·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6954098782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6954098782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−15.13546599165049, −14.41199752361975, −14.03241680858804, −13.17326964919624, −12.69964613095091, −12.38816566878904, −11.86063619673607, −11.37801031647723, −10.76410147503135, −10.18367728440150, −9.819459475852745, −9.204345264718926, −8.580516494975334, −7.740336800730682, −7.268741440333463, −6.661710178562817, −6.125623217976748, −5.677801166153095, −5.077070913872239, −4.341702927121033, −3.651409962148368, −3.027309745125812, −2.367524672871123, −1.029662714391452, −0.4092467007104108,
0.4092467007104108, 1.029662714391452, 2.367524672871123, 3.027309745125812, 3.651409962148368, 4.341702927121033, 5.077070913872239, 5.677801166153095, 6.125623217976748, 6.661710178562817, 7.268741440333463, 7.740336800730682, 8.580516494975334, 9.204345264718926, 9.819459475852745, 10.18367728440150, 10.76410147503135, 11.37801031647723, 11.86063619673607, 12.38816566878904, 12.69964613095091, 13.17326964919624, 14.03241680858804, 14.41199752361975, 15.13546599165049
