| L(s) = 1 | + 6·9-s − 6·13-s − 2·17-s + 6·25-s − 10·29-s − 6·37-s + 30·41-s − 14·49-s + 28·53-s + 10·61-s + 9·81-s + 2·101-s + 14·113-s − 36·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + 163-s + 167-s + 13·169-s + 173-s + ⋯ |
| L(s) = 1 | + 2·9-s − 1.66·13-s − 0.485·17-s + 6/5·25-s − 1.85·29-s − 0.986·37-s + 4.68·41-s − 2·49-s + 3.84·53-s + 1.28·61-s + 81-s + 0.199·101-s + 1.31·113-s − 3.32·117-s − 2·121-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 − 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.636013848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.636013848\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 7 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 + 10 T + p T^{2} )^{2}( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} ) \) |
| 43 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 67 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} )( 1 + 10 T + 11 T^{2} + 10 p T^{3} + p^{2} T^{4} ) \) |
| 97 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 227 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
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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
−7.35980518253909685575193985582, −7.11942912006529193396149136100, −7.03220731796157655742113515926, −6.80320491925983670412259406086, −6.44367094403460190356525907121, −6.23153950091752575337230252525, −5.93545879801345982654360249689, −5.62098404848355457271241564505, −5.52683977412504862741298081037, −5.16317656016863442312993040933, −4.93609722390651832568221771329, −4.73809689660409866311526501350, −4.40555019126567613805182183371, −4.12669727066899849429281092934, −4.11707234918729206463761626035, −3.74094497163267420718224581248, −3.56469633631798894588431908035, −2.90842341105189175552671471525, −2.73975812479733410530884682120, −2.53224780402548652811020075225, −2.03640995641928478005437404317, −1.88963296536091567284510408188, −1.49618214413520375105363943287, −0.78531418781299928607607743514, −0.63023673440877619986476727710,
0.63023673440877619986476727710, 0.78531418781299928607607743514, 1.49618214413520375105363943287, 1.88963296536091567284510408188, 2.03640995641928478005437404317, 2.53224780402548652811020075225, 2.73975812479733410530884682120, 2.90842341105189175552671471525, 3.56469633631798894588431908035, 3.74094497163267420718224581248, 4.11707234918729206463761626035, 4.12669727066899849429281092934, 4.40555019126567613805182183371, 4.73809689660409866311526501350, 4.93609722390651832568221771329, 5.16317656016863442312993040933, 5.52683977412504862741298081037, 5.62098404848355457271241564505, 5.93545879801345982654360249689, 6.23153950091752575337230252525, 6.44367094403460190356525907121, 6.80320491925983670412259406086, 7.03220731796157655742113515926, 7.11942912006529193396149136100, 7.35980518253909685575193985582
