| L(s) = 1 | + 2·3-s − 2·5-s + 9-s + 4·11-s − 4·15-s − 8·17-s + 8·19-s + 12·23-s + 25-s − 2·27-s − 8·29-s + 8·33-s + 4·37-s + 8·41-s − 16·43-s − 2·45-s − 4·47-s − 16·51-s + 12·53-s − 8·55-s + 16·57-s + 8·67-s + 24·69-s − 8·71-s − 8·73-s + 2·75-s + 20·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s + 1/3·9-s + 1.20·11-s − 1.03·15-s − 1.94·17-s + 1.83·19-s + 2.50·23-s + 1/5·25-s − 0.384·27-s − 1.48·29-s + 1.39·33-s + 0.657·37-s + 1.24·41-s − 2.43·43-s − 0.298·45-s − 0.583·47-s − 2.24·51-s + 1.64·53-s − 1.07·55-s + 2.11·57-s + 0.977·67-s + 2.88·69-s − 0.949·71-s − 0.936·73-s + 0.230·75-s + 2.25·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.515271173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.515271173\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 8 T + 22 T^{2} + 64 T^{3} + 387 T^{4} + 64 p T^{5} + 22 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T + 12 T^{2} - 112 T^{3} + 1127 T^{4} - 112 p T^{5} + 12 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 12 T + 64 T^{2} - 408 T^{3} + 2559 T^{4} - 408 p T^{5} + 64 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - 44 T^{2} + 975 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 4 T + 10 T^{2} + 272 T^{3} - 1925 T^{4} + 272 p T^{5} + 10 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 4 T - 10 T^{2} - 272 T^{3} - 2285 T^{4} - 272 p T^{5} - 10 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 12 T + 4 T^{2} - 408 T^{3} + 9159 T^{4} - 408 p T^{5} + 4 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^3$ | \( 1 - 110 T^{2} + 8619 T^{4} - 110 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^3$ | \( 1 - 24 T^{2} - 3145 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 8 T - 78 T^{2} - 64 T^{3} + 11387 T^{4} - 64 p T^{5} - 78 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 8 T - 26 T^{2} - 448 T^{3} - 1901 T^{4} - 448 p T^{5} - 26 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 10 T + 21 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 4 T + 162 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 4 T - 158 T^{2} - 16 T^{3} + 20931 T^{4} - 16 p T^{5} - 158 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
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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
−6.35199156963074603597301194845, −5.85673108152994803322773043909, −5.80087239793737193958514684609, −5.74744326857497927653257382528, −5.23574812180824858648451137313, −5.16616479629497580735489160022, −4.78554067123690913820549526237, −4.73545968440050448671526384938, −4.70748354416457400533999506039, −4.23891488243802816401047675971, −4.04057298159892512186933928842, −3.77053023732566682721128665302, −3.70146449682304628849505344272, −3.46091211342222305948875821695, −3.15001508716712208140466767171, −3.11559766983367245772366767450, −2.83744468405671576006946255402, −2.49090902538602759098849319268, −2.19962267147352565094812582196, −2.06291943711600544768130958689, −1.74993177782629761652323691333, −1.32279214083693940087558061571, −0.957949434699689739431536229583, −0.904469218811291555359000389187, −0.21888032036065216018290967097,
0.21888032036065216018290967097, 0.904469218811291555359000389187, 0.957949434699689739431536229583, 1.32279214083693940087558061571, 1.74993177782629761652323691333, 2.06291943711600544768130958689, 2.19962267147352565094812582196, 2.49090902538602759098849319268, 2.83744468405671576006946255402, 3.11559766983367245772366767450, 3.15001508716712208140466767171, 3.46091211342222305948875821695, 3.70146449682304628849505344272, 3.77053023732566682721128665302, 4.04057298159892512186933928842, 4.23891488243802816401047675971, 4.70748354416457400533999506039, 4.73545968440050448671526384938, 4.78554067123690913820549526237, 5.16616479629497580735489160022, 5.23574812180824858648451137313, 5.74744326857497927653257382528, 5.80087239793737193958514684609, 5.85673108152994803322773043909, 6.35199156963074603597301194845
