| L(s) = 1 | − 4·3-s − 4·7-s + 8·9-s + 2·13-s + 10·17-s + 8·19-s + 16·21-s − 4·23-s − 12·27-s − 2·37-s − 8·39-s + 12·43-s + 4·47-s + 8·49-s − 40·51-s + 14·53-s − 32·57-s + 8·59-s − 8·61-s − 32·63-s + 20·67-s + 16·69-s + 6·73-s + 32·79-s + 23·81-s + 4·83-s − 8·91-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.51·7-s + 8/3·9-s + 0.554·13-s + 2.42·17-s + 1.83·19-s + 3.49·21-s − 0.834·23-s − 2.30·27-s − 0.328·37-s − 1.28·39-s + 1.82·43-s + 0.583·47-s + 8/7·49-s − 5.60·51-s + 1.92·53-s − 4.23·57-s + 1.04·59-s − 1.02·61-s − 4.03·63-s + 2.44·67-s + 1.92·69-s + 0.702·73-s + 3.60·79-s + 23/9·81-s + 0.439·83-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8317099291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8317099291\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.29261216253602200569395796978, −10.29114244288198554202836159454, −9.751561531843854565132887073834, −9.465245993447043182873358665702, −9.056792251200399634530463683178, −8.206740596615492220000420459678, −7.56452075924571639174983515610, −7.54254681243446590685683358450, −6.77680705173972987379947169767, −6.44681744869604256638195753490, −5.94787183488346952596562801812, −5.69385524670392918644676982756, −5.28655135622173376336904391238, −5.08995377658841635889391485991, −3.98157943307833483926616460479, −3.66404240530070806974758690128, −3.22293548106315928613307069977, −2.26682324829930236713447204304, −0.875958458013331180292750983529, −0.798051517344053497583917184647,
0.798051517344053497583917184647, 0.875958458013331180292750983529, 2.26682324829930236713447204304, 3.22293548106315928613307069977, 3.66404240530070806974758690128, 3.98157943307833483926616460479, 5.08995377658841635889391485991, 5.28655135622173376336904391238, 5.69385524670392918644676982756, 5.94787183488346952596562801812, 6.44681744869604256638195753490, 6.77680705173972987379947169767, 7.54254681243446590685683358450, 7.56452075924571639174983515610, 8.206740596615492220000420459678, 9.056792251200399634530463683178, 9.465245993447043182873358665702, 9.751561531843854565132887073834, 10.29114244288198554202836159454, 10.29261216253602200569395796978
