L(s) = 1 | + 2·3-s − 4·7-s + 2·9-s + 12·13-s − 8·21-s + 6·27-s − 16·31-s + 12·37-s + 24·39-s − 12·43-s + 8·49-s − 24·61-s − 8·63-s − 4·67-s − 4·73-s + 11·81-s − 48·91-s − 32·93-s − 36·97-s + 4·103-s + 24·111-s + 24·117-s + 4·121-s + 127-s − 24·129-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1.51·7-s + 2/3·9-s + 3.32·13-s − 1.74·21-s + 1.15·27-s − 2.87·31-s + 1.97·37-s + 3.84·39-s − 1.82·43-s + 8/7·49-s − 3.07·61-s − 1.00·63-s − 0.488·67-s − 0.468·73-s + 11/9·81-s − 5.03·91-s − 3.31·93-s − 3.65·97-s + 0.394·103-s + 2.27·111-s + 2.21·117-s + 4/11·121-s + 0.0887·127-s − 2.11·129-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.166063009\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.166063009\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
good | 7 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 2 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 238 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 53 | $C_2^3$ | \( 1 + 3598 T^{4} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 - 122 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )( 1 + 22 T + 242 T^{2} + 22 p T^{3} + p^{2} T^{4} ) \) |
| 89 | $C_2^2$ | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.74081194355476005951239446274, −6.70682723394328608662238404132, −6.63822322892982365969033386983, −6.38830753419234978313566261659, −6.07640172491523132574067252375, −5.98507161111956190832112557533, −5.64292941542219610379464720818, −5.46573945500311285087098978571, −5.37523060352963320212880066359, −4.92224739812581724141635039879, −4.51578265455237073412710308412, −4.25771374398461497215379506851, −4.15613570648461564556955649550, −3.88023605353845044439284604215, −3.59706476093353242073931337957, −3.37747675322557193922622026478, −3.24418660808167617612510320903, −2.99003733534455257531375551187, −2.75488827550878834490373029855, −2.44212347839885635199971275619, −1.91715973880249491721063757355, −1.56212409570734491196575380919, −1.39990167922062572248461429956, −1.06605256918543693404033651817, −0.27657235904198068197640294691,
0.27657235904198068197640294691, 1.06605256918543693404033651817, 1.39990167922062572248461429956, 1.56212409570734491196575380919, 1.91715973880249491721063757355, 2.44212347839885635199971275619, 2.75488827550878834490373029855, 2.99003733534455257531375551187, 3.24418660808167617612510320903, 3.37747675322557193922622026478, 3.59706476093353242073931337957, 3.88023605353845044439284604215, 4.15613570648461564556955649550, 4.25771374398461497215379506851, 4.51578265455237073412710308412, 4.92224739812581724141635039879, 5.37523060352963320212880066359, 5.46573945500311285087098978571, 5.64292941542219610379464720818, 5.98507161111956190832112557533, 6.07640172491523132574067252375, 6.38830753419234978313566261659, 6.63822322892982365969033386983, 6.70682723394328608662238404132, 6.74081194355476005951239446274