| L(s) = 1 | − 4·5-s − 6·11-s + 2·13-s − 4·17-s + 19-s + 6·23-s + 11·25-s − 10·29-s − 8·31-s − 10·37-s + 6·41-s + 4·43-s + 6·47-s − 7·49-s − 2·53-s + 24·55-s + 4·59-s + 10·61-s − 8·65-s + 12·67-s + 12·71-s − 6·73-s + 4·79-s + 14·83-s + 16·85-s + 6·89-s − 4·95-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 1.80·11-s + 0.554·13-s − 0.970·17-s + 0.229·19-s + 1.25·23-s + 11/5·25-s − 1.85·29-s − 1.43·31-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 0.875·47-s − 49-s − 0.274·53-s + 3.23·55-s + 0.520·59-s + 1.28·61-s − 0.992·65-s + 1.46·67-s + 1.42·71-s − 0.702·73-s + 0.450·79-s + 1.53·83-s + 1.73·85-s + 0.635·89-s − 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6606620834\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6606620834\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.679662374505764291666553249799, −7.981479629778651626376256823836, −7.40617354494472314189652357329, −6.87293868368558502668964230985, −5.49069521766990283991625214497, −4.95253782394352153845320051606, −3.89997516957904527441695023922, −3.34912405773193512400892014972, −2.23527648027974442016861600805, −0.48030949105903681731262675923,
0.48030949105903681731262675923, 2.23527648027974442016861600805, 3.34912405773193512400892014972, 3.89997516957904527441695023922, 4.95253782394352153845320051606, 5.49069521766990283991625214497, 6.87293868368558502668964230985, 7.40617354494472314189652357329, 7.981479629778651626376256823836, 8.679662374505764291666553249799
