| L(s) = 1 | + 3·5-s + 2·7-s + 6·11-s − 5·13-s + 6·17-s − 4·19-s − 6·23-s + 5·25-s + 3·29-s − 4·31-s + 6·35-s + 10·37-s − 6·41-s − 10·43-s + 7·49-s + 12·53-s + 18·55-s + 12·59-s − 5·61-s − 15·65-s + 2·67-s + 12·71-s − 2·73-s + 12·77-s − 10·79-s + 18·85-s + 6·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.755·7-s + 1.80·11-s − 1.38·13-s + 1.45·17-s − 0.917·19-s − 1.25·23-s + 25-s + 0.557·29-s − 0.718·31-s + 1.01·35-s + 1.64·37-s − 0.937·41-s − 1.52·43-s + 49-s + 1.64·53-s + 2.42·55-s + 1.56·59-s − 0.640·61-s − 1.86·65-s + 0.244·67-s + 1.42·71-s − 0.234·73-s + 1.36·77-s − 1.12·79-s + 1.95·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1679616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.648399641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.648399641\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 57 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 5 T - 36 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 10 T + 3 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−9.877865463999594368227943876935, −9.513145508856414130670156542305, −9.162759996334966976678014034875, −8.653957286510925976905740786362, −8.291946615695808680903976279477, −7.928543513752632560381582392022, −7.24754062810363543825656749735, −7.05968156440363707028324450756, −6.45656483339695475062935820303, −6.15968183716336270652544796730, −5.68008720901731402825000360580, −5.29783298288663241460452518615, −4.83794961968080808395419789308, −4.28288968474340840483283913036, −3.87342366276170379230902321836, −3.28414556135854286180528628589, −2.45428163444717510897293725720, −2.06435114104557519694871934833, −1.57573463571160337084355565395, −0.838348582054009283819936269622,
0.838348582054009283819936269622, 1.57573463571160337084355565395, 2.06435114104557519694871934833, 2.45428163444717510897293725720, 3.28414556135854286180528628589, 3.87342366276170379230902321836, 4.28288968474340840483283913036, 4.83794961968080808395419789308, 5.29783298288663241460452518615, 5.68008720901731402825000360580, 6.15968183716336270652544796730, 6.45656483339695475062935820303, 7.05968156440363707028324450756, 7.24754062810363543825656749735, 7.928543513752632560381582392022, 8.291946615695808680903976279477, 8.653957286510925976905740786362, 9.162759996334966976678014034875, 9.513145508856414130670156542305, 9.877865463999594368227943876935
