| L(s) = 1 | + 8·5-s + 16·23-s + 24·25-s − 8·29-s + 16·43-s − 16·47-s + 4·49-s + 8·53-s + 48·67-s + 16·71-s − 16·73-s + 32·97-s − 8·101-s + 128·115-s + 28·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 36·169-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 3.33·23-s + 24/5·25-s − 1.48·29-s + 2.43·43-s − 2.33·47-s + 4/7·49-s + 1.09·53-s + 5.86·67-s + 1.89·71-s − 1.87·73-s + 3.24·97-s − 0.796·101-s + 11.9·115-s + 2.54·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.76·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(16.40128215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.40128215\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 26 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 32 T^{2} + 706 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 4 T + 60 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1094 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 3670 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7330 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 70 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 4 T + 60 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 44 T^{2} - 746 T^{4} - 44 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 14326 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 71 | $D_{4}$ | \( ( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 79 | $C_4\times C_2$ | \( 1 - 164 T^{2} + 18054 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 188 T^{2} + 20566 T^{4} - 188 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 192 T^{2} + 20450 T^{4} - 192 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 16 T + 226 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.44230384417422423822698072962, −6.16905849183434429478258517947, −5.75342954271676378654892243713, −5.71092882633406978161809065941, −5.68337696325808554607420135233, −5.38890400458878353731772284274, −5.24097923808208837279425431541, −5.20815156810270848403629058594, −4.74069361002316984864468349053, −4.71622992510796016796297414721, −4.22587661591381140889934911571, −4.01228376974065599760273592070, −3.97194765791397514818502033809, −3.32120340323026275174472131742, −3.21068923856874190041442762923, −3.11341700999844826315095394977, −2.85406355455886356601623612481, −2.28848788511216843242081316202, −2.19088929907138344121106205761, −2.07904868722621480438841046470, −1.79497165810668352076306835982, −1.77452068044452813381761043006, −0.954886663720236993772733085453, −0.904655123399420313629413041588, −0.67116515437534502075938333588,
0.67116515437534502075938333588, 0.904655123399420313629413041588, 0.954886663720236993772733085453, 1.77452068044452813381761043006, 1.79497165810668352076306835982, 2.07904868722621480438841046470, 2.19088929907138344121106205761, 2.28848788511216843242081316202, 2.85406355455886356601623612481, 3.11341700999844826315095394977, 3.21068923856874190041442762923, 3.32120340323026275174472131742, 3.97194765791397514818502033809, 4.01228376974065599760273592070, 4.22587661591381140889934911571, 4.71622992510796016796297414721, 4.74069361002316984864468349053, 5.20815156810270848403629058594, 5.24097923808208837279425431541, 5.38890400458878353731772284274, 5.68337696325808554607420135233, 5.71092882633406978161809065941, 5.75342954271676378654892243713, 6.16905849183434429478258517947, 6.44230384417422423822698072962
