| L(s) = 1 | + 2·5-s − 10·17-s − 16·19-s − 25-s − 12·31-s + 21·49-s − 16·59-s + 10·61-s − 8·67-s − 16·71-s + 18·73-s + 4·79-s − 20·85-s − 32·95-s + 36·101-s + 28·103-s − 8·107-s + 37·121-s + 2·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·155-s + 157-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2.42·17-s − 3.67·19-s − 1/5·25-s − 2.15·31-s + 3·49-s − 2.08·59-s + 1.28·61-s − 0.977·67-s − 1.89·71-s + 2.10·73-s + 0.450·79-s − 2.16·85-s − 3.28·95-s + 3.58·101-s + 2.75·103-s − 0.773·107-s + 3.36·121-s + 0.178·125-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 − 1.92·155-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.286831465\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.286831465\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| good | 5 | $D_{4}$ | \( ( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 3 p T^{2} + 200 T^{4} - 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 37 T^{2} + 576 T^{4} - 37 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 350 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 5 T + 32 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 16 T^{2} - 66 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 426 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 6 T + 38 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 6206 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 85 T^{2} + 3648 T^{4} - 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 169 T^{2} + 11484 T^{4} - 169 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 100 T^{2} + 6006 T^{4} - 100 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 - 5 T + 120 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 73 | $D_{4}$ | \( ( 1 - 9 T + 92 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 2 T + 126 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 154 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.17333231775383399717901451656, −6.09805229614701810385523546310, −5.95289627906522139328631542909, −5.72697938976650771416221653942, −5.57910366018975732276592685003, −5.26651074123904024079015086202, −4.85944625468801963236117433567, −4.75018643528261297487889780066, −4.60683008674190201247236088569, −4.42166336276566601083519562996, −4.26170588316308541084313767528, −3.90094299011055597615512525627, −3.80203283844520693579824779265, −3.58944344006895952516585196945, −3.28570031160636039478597904318, −2.96247962349701155859142772982, −2.48092740852513496844548347704, −2.46486827481052905037631686302, −2.08839203693154512691743822417, −2.07572990849311684487837238816, −1.86533902966586833474346644876, −1.66459791690092712678203934137, −1.04070646736043895234988613409, −0.52120653099828986500041316356, −0.23652182474508181986311652935,
0.23652182474508181986311652935, 0.52120653099828986500041316356, 1.04070646736043895234988613409, 1.66459791690092712678203934137, 1.86533902966586833474346644876, 2.07572990849311684487837238816, 2.08839203693154512691743822417, 2.46486827481052905037631686302, 2.48092740852513496844548347704, 2.96247962349701155859142772982, 3.28570031160636039478597904318, 3.58944344006895952516585196945, 3.80203283844520693579824779265, 3.90094299011055597615512525627, 4.26170588316308541084313767528, 4.42166336276566601083519562996, 4.60683008674190201247236088569, 4.75018643528261297487889780066, 4.85944625468801963236117433567, 5.26651074123904024079015086202, 5.57910366018975732276592685003, 5.72697938976650771416221653942, 5.95289627906522139328631542909, 6.09805229614701810385523546310, 6.17333231775383399717901451656
