| L(s) = 1 | − 16·7-s + 6·9-s + 32·23-s + 40·29-s − 88·41-s + 160·43-s − 224·47-s + 60·49-s − 56·61-s − 96·63-s + 128·67-s + 27·81-s + 224·83-s + 312·89-s + 664·101-s − 16·103-s + 192·107-s + 24·109-s + 260·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 512·161-s + ⋯ |
| L(s) = 1 | − 2.28·7-s + 2/3·9-s + 1.39·23-s + 1.37·29-s − 2.14·41-s + 3.72·43-s − 4.76·47-s + 1.22·49-s − 0.918·61-s − 1.52·63-s + 1.91·67-s + 1/3·81-s + 2.69·83-s + 3.50·89-s + 6.57·101-s − 0.155·103-s + 1.79·107-s + 0.220·109-s + 2.14·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s − 3.18·161-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.671452346\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.671452346\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $D_{4}$ | \( ( 1 + 8 T + 66 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 33894 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 92 T^{2} - 17562 T^{4} - 92 p^{4} T^{6} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 700 T^{2} + 261894 T^{4} - 700 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 196 T^{2} + 159654 T^{4} - 196 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{4} \) |
| 29 | $D_{4}$ | \( ( 1 - 20 T + 1734 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1412 T^{2} + 2268678 T^{4} - 1412 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 3292 T^{2} + 3990 p^{2} T^{4} - 3292 p^{4} T^{6} + p^{8} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 44 T + 3654 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 80 T + 4098 T^{2} - 80 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 112 T + 7362 T^{2} + 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 11068 T^{2} + 46399206 T^{4} - 11068 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 462054 T^{4} - 260 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{4} \) |
| 67 | $D_{4}$ | \( ( 1 - 64 T + 7650 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3140 T^{2} + 9051462 T^{4} - 3140 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 13372 T^{2} + 90439878 T^{4} - 13372 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 12548 T^{2} + 107282310 T^{4} - 12548 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 112 T + 16866 T^{2} - 112 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 - 156 T + 21158 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 16124 T^{2} + 135775494 T^{4} - 16124 p^{4} T^{6} + p^{8} T^{8} \) |
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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
−6.23873204757847295463996355837, −6.01986417975741245982081153048, −5.87366785926158385256240358502, −5.75772906429596853625471613111, −5.17597360859730021215206466290, −5.05796510815988795006744397750, −4.79945275071777388610763271252, −4.73508683470440880673293351407, −4.68569483408334412091896833109, −4.40617396503138587727429542553, −3.77614097101527124038440888060, −3.71567483443442812154135422248, −3.56177353724615255042843855069, −3.29153982545781815230473437143, −3.13589066261415111401285967702, −3.07513168599905536186752715493, −2.77781088400691032560341197038, −2.22469153591121380068661951247, −2.04796143478210714993342759050, −1.99187275728381430045097066167, −1.56533281567059767051961838150, −1.05149590519196067965371619855, −0.73045465955474767352133465498, −0.59342385226675932963207463235, −0.34589359404282087502747467720,
0.34589359404282087502747467720, 0.59342385226675932963207463235, 0.73045465955474767352133465498, 1.05149590519196067965371619855, 1.56533281567059767051961838150, 1.99187275728381430045097066167, 2.04796143478210714993342759050, 2.22469153591121380068661951247, 2.77781088400691032560341197038, 3.07513168599905536186752715493, 3.13589066261415111401285967702, 3.29153982545781815230473437143, 3.56177353724615255042843855069, 3.71567483443442812154135422248, 3.77614097101527124038440888060, 4.40617396503138587727429542553, 4.68569483408334412091896833109, 4.73508683470440880673293351407, 4.79945275071777388610763271252, 5.05796510815988795006744397750, 5.17597360859730021215206466290, 5.75772906429596853625471613111, 5.87366785926158385256240358502, 6.01986417975741245982081153048, 6.23873204757847295463996355837
