| L(s) = 1 | − 4·13-s − 4·17-s − 20·37-s + 8·41-s + 28·53-s + 24·61-s + 28·73-s + 18·81-s + 28·97-s − 40·101-s − 36·113-s + 20·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 1.10·13-s − 0.970·17-s − 3.28·37-s + 1.24·41-s + 3.84·53-s + 3.07·61-s + 3.27·73-s + 2·81-s + 2.84·97-s − 3.98·101-s − 3.38·113-s + 1.81·121-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 + 8/13·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.574993559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.574993559\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T^{2} )^{2}( 1 + p T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 34 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 542 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 2702 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 3326 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 - 2578 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 2606 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 - 114 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.076524232436387060319510701457, −7.961382280646779127031268536552, −7.929848131919296678801541691229, −7.30543081521069108055327674977, −6.96162693771974428196196441921, −6.95309419812714374494453756692, −6.76280009442643743826026517782, −6.66677972259291115209157186356, −6.25394911848264293709998729180, −5.67162619376543875990943035541, −5.52933697376557481619108187543, −5.33152245683689873936256126597, −5.22490651224131978580782816688, −4.87066099312671197157546526753, −4.45638240086516609016931229094, −4.06516090047597539283107897927, −4.00150205838737360138833500265, −3.57838291385557204005584967938, −3.36854187209536056126018633292, −2.81302663014717021406395112426, −2.35063255051771863388721450449, −2.14897335419290936823123306326, −2.04872627318008344173601307473, −1.10780725311689390416912559032, −0.52633571713964570671911507638,
0.52633571713964570671911507638, 1.10780725311689390416912559032, 2.04872627318008344173601307473, 2.14897335419290936823123306326, 2.35063255051771863388721450449, 2.81302663014717021406395112426, 3.36854187209536056126018633292, 3.57838291385557204005584967938, 4.00150205838737360138833500265, 4.06516090047597539283107897927, 4.45638240086516609016931229094, 4.87066099312671197157546526753, 5.22490651224131978580782816688, 5.33152245683689873936256126597, 5.52933697376557481619108187543, 5.67162619376543875990943035541, 6.25394911848264293709998729180, 6.66677972259291115209157186356, 6.76280009442643743826026517782, 6.95309419812714374494453756692, 6.96162693771974428196196441921, 7.30543081521069108055327674977, 7.929848131919296678801541691229, 7.961382280646779127031268536552, 8.076524232436387060319510701457
