| L(s) = 1 | + 2·4-s − 7-s − 14·13-s + 7·19-s + 5·25-s − 2·28-s + 7·31-s + 37-s + 10·43-s − 6·49-s − 28·52-s − 14·61-s − 8·64-s − 11·67-s + 7·73-s + 14·76-s + 13·79-s + 14·91-s + 28·97-s + 10·100-s + 7·103-s − 17·109-s + 11·121-s + 14·124-s + 127-s + 131-s − 7·133-s + ⋯ |
| L(s) = 1 | + 4-s − 0.377·7-s − 3.88·13-s + 1.60·19-s + 25-s − 0.377·28-s + 1.25·31-s + 0.164·37-s + 1.52·43-s − 6/7·49-s − 3.88·52-s − 1.79·61-s − 64-s − 1.34·67-s + 0.819·73-s + 1.60·76-s + 1.46·79-s + 1.46·91-s + 2.84·97-s + 100-s + 0.689·103-s − 1.62·109-s + 121-s + 1.25·124-s + 0.0887·127-s + 0.0873·131-s − 0.606·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8528974404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8528974404\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−15.10391512386323464488553043475, −14.89204274700472610334583207136, −14.19759421423389586027241204546, −13.81857004699226797923647639741, −12.75045934194220757550721874333, −12.43916690255645319183688183218, −11.76131590485914725834823505798, −11.73034383769830250715065081985, −10.57513541047567563154004230669, −10.20281678875964561378998763774, −9.438431329900591574938370924126, −9.250577143948027650400233365546, −7.70102303608271730101848054391, −7.56481020328747985759872945590, −6.95697683731745953258666267602, −6.22998923426805380952819476845, −5.06428004548084692695021768767, −4.71954527606832227016107159929, −2.95585781887631646524441704782, −2.45311402063529036694048344147,
2.45311402063529036694048344147, 2.95585781887631646524441704782, 4.71954527606832227016107159929, 5.06428004548084692695021768767, 6.22998923426805380952819476845, 6.95697683731745953258666267602, 7.56481020328747985759872945590, 7.70102303608271730101848054391, 9.250577143948027650400233365546, 9.438431329900591574938370924126, 10.20281678875964561378998763774, 10.57513541047567563154004230669, 11.73034383769830250715065081985, 11.76131590485914725834823505798, 12.43916690255645319183688183218, 12.75045934194220757550721874333, 13.81857004699226797923647639741, 14.19759421423389586027241204546, 14.89204274700472610334583207136, 15.10391512386323464488553043475
