| L(s) = 1 | + 289·5-s − 527·7-s + 1.31e4·9-s − 2.50e4·11-s + 4.24e4·17-s + 2.60e5·19-s − 1.06e6·23-s + 3.90e5·25-s − 1.52e5·35-s − 5.60e6·43-s + 3.79e6·45-s + 8.30e6·47-s + 5.76e6·49-s − 7.22e6·55-s + 1.76e7·61-s − 6.91e6·63-s − 4.38e7·73-s + 1.31e7·77-s + 1.29e8·81-s − 1.25e8·83-s + 1.22e7·85-s + 7.53e7·95-s − 3.28e8·99-s − 2.16e8·101-s − 3.09e8·115-s − 2.23e7·119-s + 2.14e8·121-s + ⋯ |
| L(s) = 1 | + 0.462·5-s − 0.219·7-s + 2·9-s − 1.70·11-s + 0.508·17-s + 2·19-s − 3.82·23-s + 25-s − 0.101·35-s − 1.63·43-s + 0.924·45-s + 1.70·47-s + 49-s − 0.789·55-s + 1.27·61-s − 0.438·63-s − 1.54·73-s + 0.374·77-s + 3·81-s − 2.64·83-s + 0.234·85-s + 0.924·95-s − 3.41·99-s − 2.07·101-s − 1.76·115-s − 0.111·119-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.713570242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.713570242\) |
| \(L(5)\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p^{4} T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 289 T - 307104 T^{2} - 289 p^{8} T^{3} + p^{16} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 527 T - 5487072 T^{2} + 527 p^{8} T^{3} + p^{16} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 25007 T + 410991168 T^{2} + 25007 p^{8} T^{3} + p^{16} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 42433 T - 5175197952 T^{2} - 42433 p^{8} T^{3} + p^{16} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 534718 T + p^{8} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 5602127 T + 19695626646528 T^{2} + 5602127 p^{8} T^{3} + p^{16} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 8302513 T + 45120435453408 T^{2} - 8302513 p^{8} T^{3} + p^{16} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 17661793 T + 120231618977568 T^{2} - 17661793 p^{8} T^{3} + p^{16} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 43864607 T + 1117643655370368 T^{2} + 43864607 p^{8} T^{3} + p^{16} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 62676958 T + p^{8} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p^{4} T )^{2}( 1 + p^{4} T )^{2} \) |
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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
−13.15661753246894371313363489917, −12.57968257101423554861239725267, −12.13019400027660406739594182459, −11.68070805781651558335516233744, −10.44413027928495313936639248898, −10.38195181802115331555079238348, −9.763081771393635613660181835962, −9.630426448551985718468684271327, −8.371248207549971515603366840520, −7.899911509122363414574628973538, −7.31554004764111922597798281504, −6.87748359985426873477725532188, −5.67749247403503850159854472469, −5.57675603963569393226242515498, −4.52537541861797999287531959346, −3.94054094884170219296393889532, −3.01233196809880237863100525165, −2.11723774974709751613641528380, −1.45466480341115996875250150999, −0.52169880472405629034231812918,
0.52169880472405629034231812918, 1.45466480341115996875250150999, 2.11723774974709751613641528380, 3.01233196809880237863100525165, 3.94054094884170219296393889532, 4.52537541861797999287531959346, 5.57675603963569393226242515498, 5.67749247403503850159854472469, 6.87748359985426873477725532188, 7.31554004764111922597798281504, 7.899911509122363414574628973538, 8.371248207549971515603366840520, 9.630426448551985718468684271327, 9.763081771393635613660181835962, 10.38195181802115331555079238348, 10.44413027928495313936639248898, 11.68070805781651558335516233744, 12.13019400027660406739594182459, 12.57968257101423554861239725267, 13.15661753246894371313363489917
