| L(s) = 1 | − 2·5-s − 5·11-s + 3·13-s − 17-s − 2·19-s − 8·23-s − 25-s + 6·29-s − 4·31-s + 8·37-s + 6·41-s − 4·43-s + 10·47-s − 9·53-s + 10·55-s − 4·59-s − 4·61-s − 6·65-s + 10·67-s − 5·71-s − 2·73-s + 79-s + 12·83-s + 2·85-s + 9·89-s + 4·95-s − 4·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.50·11-s + 0.832·13-s − 0.242·17-s − 0.458·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 0.718·31-s + 1.31·37-s + 0.937·41-s − 0.609·43-s + 1.45·47-s − 1.23·53-s + 1.34·55-s − 0.520·59-s − 0.512·61-s − 0.744·65-s + 1.22·67-s − 0.593·71-s − 0.234·73-s + 0.112·79-s + 1.31·83-s + 0.216·85-s + 0.953·89-s + 0.410·95-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6656858343\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6656858343\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
| 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
−13.39835349063017, −13.21785389257570, −12.50237001588072, −12.16579039867225, −11.68142768681768, −11.00190451987110, −10.75691266832521, −10.33411261404140, −9.639919973169003, −9.195784658335509, −8.406246341640504, −8.080271108237026, −7.805464262074718, −7.274234880942782, −6.488719714503655, −6.023471606739318, −5.596126538992108, −4.801344451348814, −4.352027120097845, −3.844143378115369, −3.270693908740595, −2.520813270630898, −2.101097006977378, −1.120481240734607, −0.2676210066337778,
0.2676210066337778, 1.120481240734607, 2.101097006977378, 2.520813270630898, 3.270693908740595, 3.844143378115369, 4.352027120097845, 4.801344451348814, 5.596126538992108, 6.023471606739318, 6.488719714503655, 7.274234880942782, 7.805464262074718, 8.080271108237026, 8.406246341640504, 9.195784658335509, 9.639919973169003, 10.33411261404140, 10.75691266832521, 11.00190451987110, 11.68142768681768, 12.16579039867225, 12.50237001588072, 13.21785389257570, 13.39835349063017
