| L(s) = 1 | − 3·3-s + 2·7-s + 6·9-s − 13-s − 6·21-s − 9·27-s − 3·29-s − 3·31-s − 8·37-s + 3·39-s + 3·41-s + 2·43-s − 11·47-s − 3·49-s − 14·53-s + 8·59-s + 4·61-s + 12·63-s + 4·67-s − 7·71-s + 9·73-s + 9·81-s − 4·83-s + 9·87-s + 2·89-s − 2·91-s + 9·93-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.755·7-s + 2·9-s − 0.277·13-s − 1.30·21-s − 1.73·27-s − 0.557·29-s − 0.538·31-s − 1.31·37-s + 0.480·39-s + 0.468·41-s + 0.304·43-s − 1.60·47-s − 3/7·49-s − 1.92·53-s + 1.04·59-s + 0.512·61-s + 1.51·63-s + 0.488·67-s − 0.830·71-s + 1.05·73-s + 81-s − 0.439·83-s + 0.964·87-s + 0.211·89-s − 0.209·91-s + 0.933·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 211600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8999782204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8999782204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 11 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.82732481424720, −12.57063843305134, −11.87656361637473, −11.57989818059086, −11.21560546156270, −10.82076245491319, −10.36973084548055, −9.804236161247565, −9.490913378415842, −8.686206991602432, −8.276320762643828, −7.503608891232912, −7.316718943552641, −6.652479663103066, −6.204574367447130, −5.747642158823935, −5.192893778009595, −4.784839244884510, −4.535699799554157, −3.652064656866985, −3.237065305591014, −2.040637519694272, −1.808367746583560, −0.9999828745508161, −0.3455532817004170,
0.3455532817004170, 0.9999828745508161, 1.808367746583560, 2.040637519694272, 3.237065305591014, 3.652064656866985, 4.535699799554157, 4.784839244884510, 5.192893778009595, 5.747642158823935, 6.204574367447130, 6.652479663103066, 7.316718943552641, 7.503608891232912, 8.276320762643828, 8.686206991602432, 9.490913378415842, 9.804236161247565, 10.36973084548055, 10.82076245491319, 11.21560546156270, 11.57989818059086, 11.87656361637473, 12.57063843305134, 12.82732481424720
