| L(s) = 1 | + 10·9-s − 40·13-s + 64·19-s + 56·25-s − 128·31-s − 80·37-s − 80·43-s + 14·49-s + 56·61-s − 240·67-s + 120·73-s − 128·79-s + 19·81-s − 360·97-s − 160·103-s + 72·109-s − 400·117-s + 272·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 10/9·9-s − 3.07·13-s + 3.36·19-s + 2.23·25-s − 4.12·31-s − 2.16·37-s − 1.86·43-s + 2/7·49-s + 0.918·61-s − 3.58·67-s + 1.64·73-s − 1.62·79-s + 0.234·81-s − 3.71·97-s − 1.55·103-s + 0.660·109-s − 3.41·117-s + 2.24·121-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 + 0.00613·163-s + 0.00598·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4567405083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4567405083\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 10 T^{2} + p^{4} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 56 T^{2} + 1586 T^{4} - 56 p^{4} T^{6} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 272 T^{2} + 36578 T^{4} - 272 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 20 T + 326 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 440 T^{2} + 204242 T^{4} - 440 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 + 688 T^{2} + 666818 T^{4} + 688 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1652 T^{2} + 1917638 T^{4} - 1652 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 64 T + 2246 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 20 T + p^{2} T^{2} )^{4} \) |
| 41 | $D_4\times C_2$ | \( 1 - 856 T^{2} - 2330094 T^{4} - 856 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 40 T + 3090 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 4346 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 3026 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 10532 T^{2} + 49098278 T^{4} - 10532 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 28 T + 4838 T^{2} - 28 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 120 T + 12466 T^{2} + 120 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12176 T^{2} + 74167106 T^{4} - 12176 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 60 T + 9766 T^{2} - 60 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 64 T + 10706 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 6356 T^{2} - 6983674 T^{4} - 6356 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 27832 T^{2} + 318232338 T^{4} - 27832 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 180 T + 26470 T^{2} + 180 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.85956405780597182691857646091, −6.61757445501993412963975629723, −6.41277195366614589837326064123, −5.75150625485481699378560312337, −5.60931674415537695236430113644, −5.50019875670847872288061058983, −5.36483219958968945619495808919, −4.97921362458184043534002620827, −4.97204016114713753691591037913, −4.74791412775759246426489557081, −4.70114865846714825247699994873, −3.96972514439041499882449615168, −3.87973272618361323104634838217, −3.78747433559934356180451434076, −3.34719783593049662777193052859, −3.10774058690798823714467336966, −2.90010849689404334582422554105, −2.61692658423419685085563828300, −2.48075371278524351111877364583, −1.80762041642854693768850607349, −1.68902544908545276823028148913, −1.32837603725862881117031409016, −1.30871582720775857834218977091, −0.50473135514626606948309175451, −0.11389118795495458645052159688,
0.11389118795495458645052159688, 0.50473135514626606948309175451, 1.30871582720775857834218977091, 1.32837603725862881117031409016, 1.68902544908545276823028148913, 1.80762041642854693768850607349, 2.48075371278524351111877364583, 2.61692658423419685085563828300, 2.90010849689404334582422554105, 3.10774058690798823714467336966, 3.34719783593049662777193052859, 3.78747433559934356180451434076, 3.87973272618361323104634838217, 3.96972514439041499882449615168, 4.70114865846714825247699994873, 4.74791412775759246426489557081, 4.97204016114713753691591037913, 4.97921362458184043534002620827, 5.36483219958968945619495808919, 5.50019875670847872288061058983, 5.60931674415537695236430113644, 5.75150625485481699378560312337, 6.41277195366614589837326064123, 6.61757445501993412963975629723, 6.85956405780597182691857646091
