| L(s) = 1 | + 2·2-s + 4·5-s − 8·8-s + 8·10-s + 4·13-s − 16·16-s − 20·17-s − 38·25-s + 8·26-s + 52·29-s − 40·34-s + 52·37-s − 32·40-s − 116·41-s + 50·49-s − 76·50-s + 148·53-s + 104·58-s + 52·61-s + 64·64-s + 16·65-s − 92·73-s + 104·74-s − 64·80-s − 232·82-s − 80·85-s − 164·89-s + ⋯ |
| L(s) = 1 | + 2-s + 4/5·5-s − 8-s + 4/5·10-s + 4/13·13-s − 16-s − 1.17·17-s − 1.51·25-s + 4/13·26-s + 1.79·29-s − 1.17·34-s + 1.40·37-s − 4/5·40-s − 2.82·41-s + 1.02·49-s − 1.51·50-s + 2.79·53-s + 1.79·58-s + 0.852·61-s + 64-s + 0.246·65-s − 1.26·73-s + 1.40·74-s − 4/5·80-s − 2.82·82-s − 0.941·85-s − 1.84·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.581054555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.581054555\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 194 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1874 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1346 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 382 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 74 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1150 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8930 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1390 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 11426 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{2} \) |
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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
−16.46401896900918195063056214259, −15.69281492557054453653342039559, −15.28023555956808773268952114204, −14.74388229146239206075388990351, −13.88363228384763685657306666338, −13.45308087003564286452160478962, −13.42373903787653112614499162861, −12.43004054686086872016038275790, −11.80407577388763979492934445739, −11.31681003343307050804738487324, −10.12537533857628692118880300812, −9.916523791403926152757733473869, −8.774496596680792852849204754128, −8.489478930234701161067363574037, −7.10787343481272725408843793702, −6.28912218754223913325025705155, −5.68373852429476125954772253951, −4.73935364930487997163253497663, −3.84845937326100545215129282723, −2.44719771968788722276449401148,
2.44719771968788722276449401148, 3.84845937326100545215129282723, 4.73935364930487997163253497663, 5.68373852429476125954772253951, 6.28912218754223913325025705155, 7.10787343481272725408843793702, 8.489478930234701161067363574037, 8.774496596680792852849204754128, 9.916523791403926152757733473869, 10.12537533857628692118880300812, 11.31681003343307050804738487324, 11.80407577388763979492934445739, 12.43004054686086872016038275790, 13.42373903787653112614499162861, 13.45308087003564286452160478962, 13.88363228384763685657306666338, 14.74388229146239206075388990351, 15.28023555956808773268952114204, 15.69281492557054453653342039559, 16.46401896900918195063056214259
