| L(s) = 1 | + 4·5-s − 8·17-s + 10·25-s − 8·37-s − 32·41-s − 24·43-s − 16·47-s + 16·59-s − 16·67-s + 8·79-s − 16·83-s − 32·85-s − 24·89-s − 16·101-s − 24·109-s − 20·121-s + 20·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 28·169-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 1.94·17-s + 2·25-s − 1.31·37-s − 4.99·41-s − 3.65·43-s − 2.33·47-s + 2.08·59-s − 1.95·67-s + 0.900·79-s − 1.75·83-s − 3.47·85-s − 2.54·89-s − 1.59·101-s − 2.29·109-s − 1.81·121-s + 1.78·125-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.15·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_4\times C_2$ | \( 1 + 20 T^{2} + 262 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 28 T^{2} + 454 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 8 T^{2} + 22 p T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 56 T^{2} + 1522 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 20 T^{2} + 502 T^{4} + 20 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_2^2$ | \( ( 1 + 12 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_4$ | \( ( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 48 T^{2} + 3314 T^{4} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 160 T^{2} + 13522 T^{4} + 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 + 228 T^{2} + 22358 T^{4} + 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 + 172 T^{2} + 16054 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 4 T + 82 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 8 T + 2 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 97 | $D_4\times C_2$ | \( 1 + 172 T^{2} + 19734 T^{4} + 172 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.87223001202208350685576052337, −5.55138974404603287268474298473, −5.27523024594791234156952574879, −5.21752522714593948111265809542, −5.12204563754807290726132523536, −4.97414488099601539285613514786, −4.84245327660361332400291039337, −4.60397870399126782221842931566, −4.48061127292561210279486246984, −4.16539282218736021666049446609, −3.84100570863268182893646737404, −3.70042507837242583012471022759, −3.59296885181886705669666985501, −3.22817273620760739238122135566, −3.18850676089057798262990294660, −3.07354911144017611612742271580, −2.67956189668788373395263493088, −2.44906285919382196333485093362, −2.15796601880929109716944467484, −2.12808763752692326823452323389, −1.97775226833603758850958942030, −1.45779738903558511372901745290, −1.36146200528903086116603298555, −1.32735689499523525965069715426, −1.27396874004792328221775550001, 0, 0, 0, 0,
1.27396874004792328221775550001, 1.32735689499523525965069715426, 1.36146200528903086116603298555, 1.45779738903558511372901745290, 1.97775226833603758850958942030, 2.12808763752692326823452323389, 2.15796601880929109716944467484, 2.44906285919382196333485093362, 2.67956189668788373395263493088, 3.07354911144017611612742271580, 3.18850676089057798262990294660, 3.22817273620760739238122135566, 3.59296885181886705669666985501, 3.70042507837242583012471022759, 3.84100570863268182893646737404, 4.16539282218736021666049446609, 4.48061127292561210279486246984, 4.60397870399126782221842931566, 4.84245327660361332400291039337, 4.97414488099601539285613514786, 5.12204563754807290726132523536, 5.21752522714593948111265809542, 5.27523024594791234156952574879, 5.55138974404603287268474298473, 5.87223001202208350685576052337
