L(s) = 1 | + 2·3-s − 2·4-s + 12·7-s + 2·9-s − 4·12-s − 12·13-s + 3·16-s − 6·17-s + 8·19-s + 24·21-s + 25-s − 4·27-s − 24·28-s + 4·31-s − 4·36-s + 12·37-s − 24·39-s − 12·41-s − 10·43-s − 24·47-s + 6·48-s + 62·49-s − 12·51-s + 24·52-s + 12·53-s + 16·57-s − 12·59-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 4-s + 4.53·7-s + 2/3·9-s − 1.15·12-s − 3.32·13-s + 3/4·16-s − 1.45·17-s + 1.83·19-s + 5.23·21-s + 1/5·25-s − 0.769·27-s − 4.53·28-s + 0.718·31-s − 2/3·36-s + 1.97·37-s − 3.84·39-s − 1.87·41-s − 1.52·43-s − 3.50·47-s + 0.866·48-s + 62/7·49-s − 1.68·51-s + 3.32·52-s + 1.64·53-s + 2.11·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.918064243\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.918064243\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + p T^{2} )^{2}( 1 - 5 T^{2} + p^{2} T^{4} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 6 T + 18 T^{2} + 96 T^{3} + 511 T^{4} + 96 p T^{5} + 18 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 482 T^{4} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 2730 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 444 T^{3} + 2722 T^{4} + 444 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 50 T^{2} + 60 T^{3} - 1297 T^{4} + 60 p T^{5} + 50 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 119 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 - 70 T^{2} + p^{2} T^{4} ) \) |
| 59 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 660 T^{3} + 6034 T^{4} + 660 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 67 | $D_4\times C_2$ | \( 1 - 20 T^{2} - 822 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 18 T + 162 T^{2} + 1908 T^{3} + 21247 T^{4} + 1908 p T^{5} + 162 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 11418 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6 T + 164 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12 T + 72 T^{2} + 1020 T^{3} + 14434 T^{4} + 1020 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 27258 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.910454854822502961071666982127, −9.507378864288517136660624141310, −9.066063303693335511715761406309, −9.024943728555946670271473252843, −8.622471267111808611510759585970, −8.261048674711215591901401423258, −8.058697020141944729999862625093, −7.898377072013147318414054489883, −7.68959152861224254696584351895, −7.63467490674638925596870736527, −7.15597017497016994118066883267, −6.72835072096288243517597233164, −6.49379453968045549775435707444, −5.51601220307747301594781447428, −5.25376304505927525895188092657, −5.22596456567608498812029668333, −4.72096310626181716814485448599, −4.61301069285376201831877606548, −4.57714053866190848076928242752, −4.11231289958349244744659309402, −3.13664288048129297155868845406, −3.02823978833235987402365761481, −2.12069995446381468213987645657, −1.93313057832796320205704115891, −1.52685583126304549845630754145,
1.52685583126304549845630754145, 1.93313057832796320205704115891, 2.12069995446381468213987645657, 3.02823978833235987402365761481, 3.13664288048129297155868845406, 4.11231289958349244744659309402, 4.57714053866190848076928242752, 4.61301069285376201831877606548, 4.72096310626181716814485448599, 5.22596456567608498812029668333, 5.25376304505927525895188092657, 5.51601220307747301594781447428, 6.49379453968045549775435707444, 6.72835072096288243517597233164, 7.15597017497016994118066883267, 7.63467490674638925596870736527, 7.68959152861224254696584351895, 7.898377072013147318414054489883, 8.058697020141944729999862625093, 8.261048674711215591901401423258, 8.622471267111808611510759585970, 9.024943728555946670271473252843, 9.066063303693335511715761406309, 9.507378864288517136660624141310, 9.910454854822502961071666982127