L(s) = 1 | − 2·3-s − 2·4-s − 2·5-s − 4·7-s − 9-s − 2·11-s + 4·12-s − 4·13-s + 4·15-s + 4·17-s − 2·19-s + 4·20-s + 8·21-s − 6·23-s − 7·25-s + 6·27-s + 8·28-s − 4·29-s − 10·31-s + 4·33-s + 8·35-s + 2·36-s + 6·37-s + 8·39-s + 8·41-s + 12·43-s + 4·44-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 4-s − 0.894·5-s − 1.51·7-s − 1/3·9-s − 0.603·11-s + 1.15·12-s − 1.10·13-s + 1.03·15-s + 0.970·17-s − 0.458·19-s + 0.894·20-s + 1.74·21-s − 1.25·23-s − 7/5·25-s + 1.15·27-s + 1.51·28-s − 0.742·29-s − 1.79·31-s + 0.696·33-s + 1.35·35-s + 1/3·36-s + 0.986·37-s + 1.28·39-s + 1.24·41-s + 1.82·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 3 | $D_{4}$ | \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 16 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 36 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 10 T + 85 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 33 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 122 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6 T + 125 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 88 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 18 T + 213 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 22 T + 261 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 8 T + 90 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 32 T + 412 T^{2} + 32 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 168 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 105 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 193 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.07190025498865328151400289003, −11.78528494648351285918908043889, −11.27778035721523365343604018422, −10.67603568548032238096009848013, −10.17657948352155952925893447062, −9.760607320506683258382618585141, −9.121411903582605984216326523335, −8.961927499800109227389829265358, −7.942479983460655289164991279617, −7.46822210305757932839889624254, −7.29991824030977858611988412521, −5.99919994480041418957486446423, −5.85127016788906070010672256042, −5.58376600789641197137620065565, −4.37512198625136697918539487478, −4.22551461551423115358213443029, −3.33673779054846715421052009264, −2.48727755494252447957045829685, 0, 0,
2.48727755494252447957045829685, 3.33673779054846715421052009264, 4.22551461551423115358213443029, 4.37512198625136697918539487478, 5.58376600789641197137620065565, 5.85127016788906070010672256042, 5.99919994480041418957486446423, 7.29991824030977858611988412521, 7.46822210305757932839889624254, 7.942479983460655289164991279617, 8.961927499800109227389829265358, 9.121411903582605984216326523335, 9.760607320506683258382618585141, 10.17657948352155952925893447062, 10.67603568548032238096009848013, 11.27778035721523365343604018422, 11.78528494648351285918908043889, 12.07190025498865328151400289003