L(s) = 1 | + 8·25-s − 28·49-s − 40·61-s + 80·109-s − 44·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
L(s) = 1 | + 8/5·25-s − 4·49-s − 5.12·61-s + 7.66·109-s − 4·121-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.53·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.819353097\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.819353097\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 16 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + 40 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 80 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 56 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + 160 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
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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
−6.13494086915020075409865432240, −6.06546148494935167464680147882, −5.97818947518039396037975198588, −5.72787517604859396369461257241, −5.31939642924819921024317444417, −5.03048874813687618606417851831, −4.90272712043525097576874807814, −4.83514446760996940854890576266, −4.62503756267678159935323704540, −4.46622408834133961545820480082, −4.30779398838333144128922440316, −3.78018416011000616958745541202, −3.57748127449401638057062749088, −3.44829073487706481654802646732, −3.21814698933365120573836652954, −3.08063301660503382020800422036, −2.73911539099751124753203876597, −2.60226033821790172229266870186, −2.12282744702941616672039386016, −2.04659483141336739451335690868, −1.42939919657847700598358526782, −1.37816120288386341083677580056, −1.34541006016027690662371859658, −0.55989771367511004188519884012, −0.25854935414654121881177736660,
0.25854935414654121881177736660, 0.55989771367511004188519884012, 1.34541006016027690662371859658, 1.37816120288386341083677580056, 1.42939919657847700598358526782, 2.04659483141336739451335690868, 2.12282744702941616672039386016, 2.60226033821790172229266870186, 2.73911539099751124753203876597, 3.08063301660503382020800422036, 3.21814698933365120573836652954, 3.44829073487706481654802646732, 3.57748127449401638057062749088, 3.78018416011000616958745541202, 4.30779398838333144128922440316, 4.46622408834133961545820480082, 4.62503756267678159935323704540, 4.83514446760996940854890576266, 4.90272712043525097576874807814, 5.03048874813687618606417851831, 5.31939642924819921024317444417, 5.72787517604859396369461257241, 5.97818947518039396037975198588, 6.06546148494935167464680147882, 6.13494086915020075409865432240