| L(s) = 1 | + 32·13-s + 16·47-s + 64·89-s − 80·101-s + 32·103-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 504·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | + 8.87·13-s + 2.33·47-s + 6.78·89-s − 7.96·101-s + 3.15·103-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 + 38.7·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 17^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(23.90042328\) |
| \(L(\frac12)\) |
\(\approx\) |
\(23.90042328\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( ( 1 - 8 T^{2} + 32 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )( 1 + 8 T^{2} + 32 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 5 | \( 1 + 32 T^{4} + 994 T^{8} + 32 p^{4} T^{12} + p^{8} T^{16} \) |
| 7 | \( 1 + 96 T^{4} + 5954 T^{8} + 96 p^{4} T^{12} + p^{8} T^{16} \) |
| 11 | \( 1 - 192 T^{4} + 37346 T^{8} - 192 p^{4} T^{12} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 8 T + 34 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 40 T^{2} + 994 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 160 T^{4} - 358718 T^{8} - 160 p^{4} T^{12} + p^{8} T^{16} \) |
| 29 | \( 1 - 1120 T^{4} + 1326754 T^{8} - 1120 p^{4} T^{12} + p^{8} T^{16} \) |
| 31 | \( 1 - 1440 T^{4} + 1756034 T^{8} - 1440 p^{4} T^{12} + p^{8} T^{16} \) |
| 37 | \( 1 - 2080 T^{4} + 4202722 T^{8} - 2080 p^{4} T^{12} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 96 T^{2} + 4608 T^{4} - 96 p^{2} T^{6} + p^{4} T^{8} )( 1 + 96 T^{2} + 4608 T^{4} + 96 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 43 | \( ( 1 - 68 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + 16 T^{2} - 590 T^{4} + 16 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( ( 1 - 8 T^{2} + 706 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( 1 + 13920 T^{4} + 76018082 T^{8} + 13920 p^{4} T^{12} + p^{8} T^{16} \) |
| 67 | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( 1 + 5280 T^{4} + 8430914 T^{8} + 5280 p^{4} T^{12} + p^{8} T^{16} \) |
| 73 | \( 1 - 1728 T^{4} + 36235586 T^{8} - 1728 p^{4} T^{12} + p^{8} T^{16} \) |
| 79 | \( 1 - 18400 T^{4} + 156288514 T^{8} - 18400 p^{4} T^{12} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 264 T^{2} + 30050 T^{4} - 264 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 16 T + 234 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{4} \) |
| 97 | \( 1 - 128 T^{4} - 166366974 T^{8} - 128 p^{4} T^{12} + p^{8} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.09919369792981757020999418097, −4.01083824992783294469970959124, −3.82538939150805666439247309599, −3.76739067874021526740870715616, −3.66623262009424204171984845026, −3.61067796929851739623713204229, −3.42189054894930003508327049584, −3.41635567601263042976619417581, −3.23237752390100024347365344970, −3.13596519771280408179986942851, −2.95919165467275265524655641373, −2.72018602179863458520852867786, −2.42719732488744664151537159235, −2.37999249425223747659195787855, −2.32429635579828639199755985322, −2.10295901844408893718400149016, −1.74631632658931603400040455832, −1.47766695776091978470543865316, −1.47244879970372392807426613690, −1.46771171713700273771005570251, −1.24155262878111074657431470916, −0.976594090218549894210275925859, −0.959998233088179089306311071610, −0.65295228600343902833099375684, −0.44358918079803255356454725316,
0.44358918079803255356454725316, 0.65295228600343902833099375684, 0.959998233088179089306311071610, 0.976594090218549894210275925859, 1.24155262878111074657431470916, 1.46771171713700273771005570251, 1.47244879970372392807426613690, 1.47766695776091978470543865316, 1.74631632658931603400040455832, 2.10295901844408893718400149016, 2.32429635579828639199755985322, 2.37999249425223747659195787855, 2.42719732488744664151537159235, 2.72018602179863458520852867786, 2.95919165467275265524655641373, 3.13596519771280408179986942851, 3.23237752390100024347365344970, 3.41635567601263042976619417581, 3.42189054894930003508327049584, 3.61067796929851739623713204229, 3.66623262009424204171984845026, 3.76739067874021526740870715616, 3.82538939150805666439247309599, 4.01083824992783294469970959124, 4.09919369792981757020999418097
Plot not available for L-functions of degree greater than 10.
