| L(s) = 1 | + 9·9-s + 10·11-s + 6·19-s − 4·29-s − 16·31-s + 24·41-s + 25·49-s + 32·59-s + 18·61-s + 2·71-s − 52·79-s + 44·81-s − 26·89-s + 90·99-s − 16·101-s + 22·109-s + 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 34·169-s + ⋯ |
| L(s) = 1 | + 3·9-s + 3.01·11-s + 1.37·19-s − 0.742·29-s − 2.87·31-s + 3.74·41-s + 25/7·49-s + 4.16·59-s + 2.30·61-s + 0.237·71-s − 5.85·79-s + 44/9·81-s − 2.75·89-s + 9.04·99-s − 1.59·101-s + 2.10·109-s + 1.90·121-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 + 2.61·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(13.31290143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.31290143\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - p^{2} T^{2} + 37 T^{4} - p^{4} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 253 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 667 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 3 T + 39 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 2853 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2 T + 14 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 58 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 90 T^{2} + 4043 T^{4} - 90 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 12 T + 98 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 41 T^{2} + 1837 T^{4} - 41 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 16 T + 3 p T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 9 T + 81 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 10 T^{2} + 7003 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - T + 111 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 229 T^{2} + 23617 T^{4} - 229 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{4} \) |
| 83 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 13558 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 13 T + 219 T^{2} + 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 115 T^{2} + 11313 T^{4} + 115 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.53603946395755069195005591377, −5.96230717351104819861041500264, −5.79513246825269741848861240989, −5.77183554536342898330701651165, −5.64415778349626668303193396116, −5.29528000484163959099731291304, −5.20063399667294723698147902263, −4.73653743544316360599099140270, −4.63396862927757360501259882633, −4.06820018560608554340896001726, −4.06418643539251438830989847431, −3.96994804323668625536757022247, −3.90539073371154426215315717753, −3.78976005464349541817415019704, −3.75174643198351229914560768422, −2.90926421208732648201384177879, −2.79891927563277161356798115452, −2.51910115788965853053624621621, −2.29153817861379380726177838638, −1.75736265569842592113232881297, −1.63699084149871091685059296801, −1.35115458686132719875381952762, −1.22875561702032368554407319576, −0.813357452381791073417815152508, −0.63168264772818072014805149516,
0.63168264772818072014805149516, 0.813357452381791073417815152508, 1.22875561702032368554407319576, 1.35115458686132719875381952762, 1.63699084149871091685059296801, 1.75736265569842592113232881297, 2.29153817861379380726177838638, 2.51910115788965853053624621621, 2.79891927563277161356798115452, 2.90926421208732648201384177879, 3.75174643198351229914560768422, 3.78976005464349541817415019704, 3.90539073371154426215315717753, 3.96994804323668625536757022247, 4.06418643539251438830989847431, 4.06820018560608554340896001726, 4.63396862927757360501259882633, 4.73653743544316360599099140270, 5.20063399667294723698147902263, 5.29528000484163959099731291304, 5.64415778349626668303193396116, 5.77183554536342898330701651165, 5.79513246825269741848861240989, 5.96230717351104819861041500264, 6.53603946395755069195005591377
