L(s) = 1 | − 2·4-s + 3·16-s − 2·49-s − 4·64-s − 2·81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 4·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
L(s) = 1 | − 2·4-s + 3·16-s − 2·49-s − 4·64-s − 2·81-s + 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 4·196-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2222061295\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2222061295\) |
\(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
good | 3 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−8.731615248253548933574620284630, −8.727191755292717787655958912301, −8.675369046753481445219357500126, −8.245557687611407096365883087303, −7.73410312088666485613325094090, −7.70262059668064359062657271905, −7.54135244622423929370694674942, −7.25753457813564924206943884245, −6.66647484461294080438301284079, −6.41855386499090721004024537448, −6.38754131075484860594574741791, −5.72664450290764753759245159447, −5.70606865385969620320556948495, −5.40567304635705398602777906994, −4.95575255671963414409236138270, −4.83123021718559643282327601472, −4.44019658256500854570704243426, −4.34726461970485868771277807788, −3.83008646919882056223614640972, −3.53872943663913842518785477498, −3.31518019814357706344161295430, −2.87554991319898098850264247325, −2.39305163480691963301849715955, −1.67322738982507795003024301670, −1.19131665193209874420607157325,
1.19131665193209874420607157325, 1.67322738982507795003024301670, 2.39305163480691963301849715955, 2.87554991319898098850264247325, 3.31518019814357706344161295430, 3.53872943663913842518785477498, 3.83008646919882056223614640972, 4.34726461970485868771277807788, 4.44019658256500854570704243426, 4.83123021718559643282327601472, 4.95575255671963414409236138270, 5.40567304635705398602777906994, 5.70606865385969620320556948495, 5.72664450290764753759245159447, 6.38754131075484860594574741791, 6.41855386499090721004024537448, 6.66647484461294080438301284079, 7.25753457813564924206943884245, 7.54135244622423929370694674942, 7.70262059668064359062657271905, 7.73410312088666485613325094090, 8.245557687611407096365883087303, 8.675369046753481445219357500126, 8.727191755292717787655958912301, 8.731615248253548933574620284630