| L(s) = 1 | − 6·3-s − 52·7-s − 45·9-s − 100·13-s − 716·19-s + 312·21-s + 962·25-s + 756·27-s + 1.48e3·31-s − 3.74e3·37-s + 600·39-s − 524·43-s − 2.77e3·49-s + 4.29e3·57-s + 2.97e3·61-s + 2.34e3·63-s − 8.97e3·67-s + 580·73-s − 5.77e3·75-s − 1.96e4·79-s − 891·81-s + 5.20e3·91-s − 8.90e3·93-s − 956·97-s − 4.27e3·103-s + 9.50e3·109-s + 2.24e4·111-s + ⋯ |
| L(s) = 1 | − 2/3·3-s − 1.06·7-s − 5/9·9-s − 0.591·13-s − 1.98·19-s + 0.707·21-s + 1.53·25-s + 1.03·27-s + 1.54·31-s − 2.73·37-s + 0.394·39-s − 0.283·43-s − 1.15·49-s + 1.32·57-s + 0.798·61-s + 0.589·63-s − 1.99·67-s + 0.108·73-s − 1.02·75-s − 3.14·79-s − 0.135·81-s + 0.627·91-s − 1.02·93-s − 0.101·97-s − 0.403·103-s + 0.799·109-s + 1.82·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.001503432222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.001503432222\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| 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
−12.30859688201765729085415328084, −11.49825165505652599826483018022, −11.26997783982606775962658668845, −10.37564494617027289030602064472, −10.35651499076180984816471407427, −9.870233113643588755695178256472, −8.819549258553470313927477307279, −8.790527956198353710529486003715, −8.262711002976149926893954731749, −7.25749472262113102866089653811, −6.84131001841701018238764181381, −6.29891616314741865820476665042, −6.03306532166548153872064936715, −5.00333185588954700742622876989, −4.80815587581553464301803499792, −3.85226454291005108279961763986, −3.03493503360941713895566271691, −2.53577130884130229195396356759, −1.37984805575674823181323704375, −0.01472090498294290853771712953,
0.01472090498294290853771712953, 1.37984805575674823181323704375, 2.53577130884130229195396356759, 3.03493503360941713895566271691, 3.85226454291005108279961763986, 4.80815587581553464301803499792, 5.00333185588954700742622876989, 6.03306532166548153872064936715, 6.29891616314741865820476665042, 6.84131001841701018238764181381, 7.25749472262113102866089653811, 8.262711002976149926893954731749, 8.790527956198353710529486003715, 8.819549258553470313927477307279, 9.870233113643588755695178256472, 10.35651499076180984816471407427, 10.37564494617027289030602064472, 11.26997783982606775962658668845, 11.49825165505652599826483018022, 12.30859688201765729085415328084
