| L(s) = 1 | − 6·3-s − 8·4-s + 52·7-s − 45·9-s + 48·12-s + 100·13-s + 64·16-s − 716·19-s − 312·21-s + 962·25-s + 756·27-s − 416·28-s − 1.48e3·31-s + 360·36-s + 3.74e3·37-s − 600·39-s − 524·43-s − 384·48-s − 2.77e3·49-s − 800·52-s + 4.29e3·57-s − 2.97e3·61-s − 2.34e3·63-s − 512·64-s − 8.97e3·67-s + 580·73-s − 5.77e3·75-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 1/2·4-s + 1.06·7-s − 5/9·9-s + 1/3·12-s + 0.591·13-s + 1/4·16-s − 1.98·19-s − 0.707·21-s + 1.53·25-s + 1.03·27-s − 0.530·28-s − 1.54·31-s + 5/18·36-s + 2.73·37-s − 0.394·39-s − 0.283·43-s − 1/6·48-s − 1.15·49-s − 0.295·52-s + 1.32·57-s − 0.798·61-s − 0.589·63-s − 1/8·64-s − 1.99·67-s + 0.108·73-s − 1.02·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6319552182\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6319552182\) |
| \(L(3)\) |
|
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 | $C_2$ | \( 1 + p^{3} T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 p T + p^{4} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 962 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 26 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 15170 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 50 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 125570 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 358 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 420290 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 666238 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 742 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1874 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 155710 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 262 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 6879362 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 15571010 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 20937410 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 1486 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4486 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 38122562 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 290 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 9818 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 44355074 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 64012610 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 478 T + p^{4} T^{2} )^{2} \) |
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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
−23.17024447456488056848581217564, −22.34967934401573238287634357848, −21.68275597311725115021911028949, −20.95712969799655834888736135005, −20.18943605014334686089890033214, −19.25357393362154487622157535866, −18.14684445262715483101845034384, −17.98253016469205534137148310505, −16.81785818390148454645443886698, −16.53740745237521699110690249021, −14.75892510562098282313330351567, −14.71009596683573888842759605438, −13.30418587357277222887227752585, −12.45785439001653219150492667215, −11.17254729309775689129723266706, −10.77938771145848122171120937157, −9.032356027666629672811491447864, −8.094922226128406330354448144987, −6.21402325888917114417235131314, −4.72066524241859762569471483450,
4.72066524241859762569471483450, 6.21402325888917114417235131314, 8.094922226128406330354448144987, 9.032356027666629672811491447864, 10.77938771145848122171120937157, 11.17254729309775689129723266706, 12.45785439001653219150492667215, 13.30418587357277222887227752585, 14.71009596683573888842759605438, 14.75892510562098282313330351567, 16.53740745237521699110690249021, 16.81785818390148454645443886698, 17.98253016469205534137148310505, 18.14684445262715483101845034384, 19.25357393362154487622157535866, 20.18943605014334686089890033214, 20.95712969799655834888736135005, 21.68275597311725115021911028949, 22.34967934401573238287634357848, 23.17024447456488056848581217564
