| L(s) = 1 | + 5-s − 7-s − 3·9-s + 6·11-s − 6·13-s − 4·17-s + 14·19-s + 23-s + 5·25-s − 2·29-s − 10·31-s − 35-s − 12·37-s + 8·41-s + 10·43-s − 3·45-s − 8·47-s + 4·53-s + 6·55-s − 7·61-s + 3·63-s − 6·65-s + 12·67-s + 30·71-s − 4·73-s − 6·77-s − 79-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 9-s + 1.80·11-s − 1.66·13-s − 0.970·17-s + 3.21·19-s + 0.208·23-s + 25-s − 0.371·29-s − 1.79·31-s − 0.169·35-s − 1.97·37-s + 1.24·41-s + 1.52·43-s − 0.447·45-s − 1.16·47-s + 0.549·53-s + 0.809·55-s − 0.896·61-s + 0.377·63-s − 0.744·65-s + 1.46·67-s + 3.56·71-s − 0.468·73-s − 0.683·77-s − 0.112·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 254016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.710795790\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.710795790\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14378340108373810487657285953, −10.92216905095768275382057117270, −10.02993838421158318267995573610, −9.689927047430379837963445274130, −9.359254759688915349556621399809, −9.024934753800349899135184143228, −8.790442898467280386352089049275, −7.88063406999135904968873483830, −7.38053010498683180666534694982, −7.04079806487045421841667477210, −6.70237827658543844663769146392, −5.99138620537582391937242587918, −5.43647709877168708928633998530, −5.17418723791595700847497098426, −4.57478392809827290118619777348, −3.54373699626362443910998565605, −3.42933843704614823781603007959, −2.57894530970648102149339949371, −1.87888496616004382779534560991, −0.806745403398612268029706803062,
0.806745403398612268029706803062, 1.87888496616004382779534560991, 2.57894530970648102149339949371, 3.42933843704614823781603007959, 3.54373699626362443910998565605, 4.57478392809827290118619777348, 5.17418723791595700847497098426, 5.43647709877168708928633998530, 5.99138620537582391937242587918, 6.70237827658543844663769146392, 7.04079806487045421841667477210, 7.38053010498683180666534694982, 7.88063406999135904968873483830, 8.790442898467280386352089049275, 9.024934753800349899135184143228, 9.359254759688915349556621399809, 9.689927047430379837963445274130, 10.02993838421158318267995573610, 10.92216905095768275382057117270, 11.14378340108373810487657285953
