| L(s) = 1 | + 3·9-s − 4·13-s + 14·17-s − 4·25-s + 10·29-s + 18·37-s − 6·41-s − 5·49-s + 16·53-s − 6·61-s + 9·81-s + 6·89-s + 30·97-s − 34·101-s − 10·113-s − 12·117-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 42·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 9-s − 1.10·13-s + 3.39·17-s − 4/5·25-s + 1.85·29-s + 2.95·37-s − 0.937·41-s − 5/7·49-s + 2.19·53-s − 0.768·61-s + 81-s + 0.635·89-s + 3.04·97-s − 3.38·101-s − 0.940·113-s − 1.10·117-s + 3/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 3.39·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.053796956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.053796956\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 3 T^{2} - 112 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 + 13 T^{2} - 192 T^{4} + 13 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 - 19 T^{2} - 168 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 3 T + 44 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 59 T^{2} + 1632 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 + 69 T^{2} + 1280 T^{4} + 69 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 93 T^{2} + 3608 T^{4} + 93 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 3 T + 92 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−7.85999433137825841238346958449, −7.84837922966356476336148971811, −7.67431865783985467018022019400, −7.65184186003859798424435421020, −7.09480649625657389865806807161, −7.00133026286185184359867541163, −6.55885706918457310138539892417, −6.43284603740802420437548879113, −6.18570872055557231223196832515, −5.69139142263664641777595746280, −5.51169308262128402487868568562, −5.50461241793743242696247178041, −4.99253488121029589828855222896, −4.69551890972061568647975851003, −4.55274025705988453260866714081, −4.19300724258586834100372526043, −3.77900151792917559691653252644, −3.64303988130129048774757290899, −3.13280708837345839580850573153, −2.93104039303353373560077906867, −2.54894164571844234500184399434, −2.20001653520271263739376382071, −1.57833498637778882692703617024, −1.11471873287324465725549016299, −0.797184005518751079591302016406,
0.797184005518751079591302016406, 1.11471873287324465725549016299, 1.57833498637778882692703617024, 2.20001653520271263739376382071, 2.54894164571844234500184399434, 2.93104039303353373560077906867, 3.13280708837345839580850573153, 3.64303988130129048774757290899, 3.77900151792917559691653252644, 4.19300724258586834100372526043, 4.55274025705988453260866714081, 4.69551890972061568647975851003, 4.99253488121029589828855222896, 5.50461241793743242696247178041, 5.51169308262128402487868568562, 5.69139142263664641777595746280, 6.18570872055557231223196832515, 6.43284603740802420437548879113, 6.55885706918457310138539892417, 7.00133026286185184359867541163, 7.09480649625657389865806807161, 7.65184186003859798424435421020, 7.67431865783985467018022019400, 7.84837922966356476336148971811, 7.85999433137825841238346958449
