| L(s) = 1 | − 8·13-s − 24·23-s − 2·25-s + 24·37-s − 32·47-s + 16·49-s − 16·59-s − 24·61-s − 32·71-s − 8·73-s + 16·83-s + 8·97-s + 16·107-s + 24·109-s − 40·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 24·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | − 2.21·13-s − 5.00·23-s − 2/5·25-s + 3.94·37-s − 4.66·47-s + 16/7·49-s − 2.08·59-s − 3.07·61-s − 3.79·71-s − 0.936·73-s + 1.75·83-s + 0.812·97-s + 1.54·107-s + 2.29·109-s − 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.84·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6695257998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6695257998\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $D_4\times C_2$ | \( 1 - 16 T^{2} + 130 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_{4}$ | \( ( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 20 T^{2} + 166 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_4$ | \( ( 1 + 12 T + 74 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 1094 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 12 T + 108 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 96 T^{2} + 4514 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7030 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 53 | $D_4\times C_2$ | \( 1 - 164 T^{2} + 11830 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 8 T + 132 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $C_4\times C_2$ | \( 1 - 164 T^{2} + 18054 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 8 T + 174 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 128 T^{2} + 7138 T^{4} - 128 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.91438624360802533677076126129, −6.45823156265565079586788997751, −6.21717837056333008368079013286, −5.97539100469090254141996378376, −5.95948552544200019333695024710, −5.92784283386793990284278201989, −5.80563577899458983470857701910, −5.08690864063991979921661330973, −4.92257746602033351616623703773, −4.74758819023637021587075067402, −4.65597475409727642130979728446, −4.26622178013737711065093141909, −4.19546264363826974583081647761, −3.97140019427610637223494009866, −3.69506006802044428832936526380, −3.19903527324773265687298610837, −3.01337810493385243457634222790, −2.72980270772981524649480466262, −2.65660984705249909879703872215, −2.03507861084217419073226396922, −1.99418399914460421491730417488, −1.65401665061026502054740811334, −1.49662067040026699176015477520, −0.52266399797388193265749536457, −0.23842021868497268428215856140,
0.23842021868497268428215856140, 0.52266399797388193265749536457, 1.49662067040026699176015477520, 1.65401665061026502054740811334, 1.99418399914460421491730417488, 2.03507861084217419073226396922, 2.65660984705249909879703872215, 2.72980270772981524649480466262, 3.01337810493385243457634222790, 3.19903527324773265687298610837, 3.69506006802044428832936526380, 3.97140019427610637223494009866, 4.19546264363826974583081647761, 4.26622178013737711065093141909, 4.65597475409727642130979728446, 4.74758819023637021587075067402, 4.92257746602033351616623703773, 5.08690864063991979921661330973, 5.80563577899458983470857701910, 5.92784283386793990284278201989, 5.95948552544200019333695024710, 5.97539100469090254141996378376, 6.21717837056333008368079013286, 6.45823156265565079586788997751, 6.91438624360802533677076126129
