| L(s) = 1 | + 9-s − 4·11-s − 8·19-s − 18·29-s − 16·31-s − 24·41-s + 5·49-s + 24·59-s + 26·61-s + 12·71-s − 2·79-s − 12·81-s − 6·89-s − 4·99-s − 42·101-s + 10·109-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 50·169-s + ⋯ |
| L(s) = 1 | + 1/3·9-s − 1.20·11-s − 1.83·19-s − 3.34·29-s − 2.87·31-s − 3.74·41-s + 5/7·49-s + 3.12·59-s + 3.32·61-s + 1.42·71-s − 0.225·79-s − 4/3·81-s − 0.635·89-s − 0.402·99-s − 4.17·101-s + 0.957·109-s + 0.909·121-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 + 3.84·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9548213815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9548213815\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - T^{2} + 13 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 5 T^{2} - 27 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 53 T^{2} + 1233 T^{4} - 53 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 77 T^{2} + 2493 T^{4} - 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 9 T + 73 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 8 T + 57 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 98 T^{2} + 4803 T^{4} - 98 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 74 T^{2} + 2763 T^{4} - 74 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 77 T^{2} + 3273 T^{4} - 77 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 12 T + 133 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 13 T + 159 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 118 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 6 T - 38 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 221 T^{2} + 22233 T^{4} - 221 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + T - 99 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 65 T^{2} + 13653 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 3 T + 175 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 233 T^{2} + 30873 T^{4} - 233 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.99816685704410290554520504568, −6.97071852554104483422356760541, −6.75360959831682662220650797476, −6.55553196354825656720438605706, −6.13394506985107170407321533750, −5.79907829755189068667780124479, −5.53162913909998488141051766085, −5.37280980341232620538171177257, −5.32207774160311750296445549845, −5.13668151145891497198652472129, −5.10945551890083596516048082373, −4.21864141573697743138806802541, −4.09564233582927254930854269768, −4.04929585101012358080694340617, −3.97658914887524779390489554965, −3.56177444625620851077238643058, −3.12619379043241550266004382526, −3.09275382058044285948534933225, −2.58724957592837577759758047796, −2.14914647901678809567186536410, −1.95458200588052628896219001414, −1.79352427427937454481491753910, −1.66072238751476767802702823504, −0.62501305905652533546536089243, −0.28892035120979027137618594892,
0.28892035120979027137618594892, 0.62501305905652533546536089243, 1.66072238751476767802702823504, 1.79352427427937454481491753910, 1.95458200588052628896219001414, 2.14914647901678809567186536410, 2.58724957592837577759758047796, 3.09275382058044285948534933225, 3.12619379043241550266004382526, 3.56177444625620851077238643058, 3.97658914887524779390489554965, 4.04929585101012358080694340617, 4.09564233582927254930854269768, 4.21864141573697743138806802541, 5.10945551890083596516048082373, 5.13668151145891497198652472129, 5.32207774160311750296445549845, 5.37280980341232620538171177257, 5.53162913909998488141051766085, 5.79907829755189068667780124479, 6.13394506985107170407321533750, 6.55553196354825656720438605706, 6.75360959831682662220650797476, 6.97071852554104483422356760541, 6.99816685704410290554520504568
