| L(s) = 1 | + 2·5-s − 2·7-s − 2·11-s − 2·13-s + 4·17-s − 23-s − 25-s + 10·29-s − 4·35-s − 4·37-s + 6·41-s + 4·43-s + 8·47-s − 3·49-s + 6·53-s − 4·55-s + 4·59-s + 8·61-s − 4·65-s + 4·67-s + 8·71-s + 6·73-s + 4·77-s − 6·79-s − 6·83-s + 8·85-s + 4·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 0.603·11-s − 0.554·13-s + 0.970·17-s − 0.208·23-s − 1/5·25-s + 1.85·29-s − 0.676·35-s − 0.657·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s − 0.539·55-s + 0.520·59-s + 1.02·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.455·77-s − 0.675·79-s − 0.658·83-s + 0.867·85-s + 0.423·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.895681541\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.895681541\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.691525761170489948523465951650, −7.86859273391303121876487182190, −7.09812535712644520776771770967, −6.30441142064813868830534282975, −5.64908697772968539551467121336, −4.97856939235969581739114352032, −3.88461543205964426405952635257, −2.87579689372809428393451063907, −2.20792305223147413038698650060, −0.815254458950501667922565113690,
0.815254458950501667922565113690, 2.20792305223147413038698650060, 2.87579689372809428393451063907, 3.88461543205964426405952635257, 4.97856939235969581739114352032, 5.64908697772968539551467121336, 6.30441142064813868830534282975, 7.09812535712644520776771770967, 7.86859273391303121876487182190, 8.691525761170489948523465951650
