| L(s) = 1 | + 4·2-s + 12·4-s + 8·5-s + 32·8-s + 32·10-s + 10·13-s + 80·16-s − 16·17-s + 96·20-s + 25·25-s + 40·26-s − 40·29-s + 192·32-s − 64·34-s + 70·37-s + 256·40-s − 160·41-s + 98·49-s + 100·50-s + 120·52-s − 112·53-s − 160·58-s + 22·61-s + 448·64-s + 80·65-s − 192·68-s − 110·73-s + ⋯ |
| L(s) = 1 | + 2·2-s + 3·4-s + 8/5·5-s + 4·8-s + 16/5·10-s + 0.769·13-s + 5·16-s − 0.941·17-s + 24/5·20-s + 25-s + 1.53·26-s − 1.37·29-s + 6·32-s − 1.88·34-s + 1.89·37-s + 32/5·40-s − 3.90·41-s + 2·49-s + 2·50-s + 2.30·52-s − 2.11·53-s − 2.75·58-s + 0.360·61-s + 7·64-s + 1.23·65-s − 2.82·68-s − 1.50·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(12.45574161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.45574161\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T + 39 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 10 T - 69 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 16 T - 33 T^{2} + 16 p^{2} T^{3} + p^{4} T^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40 T + 759 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 70 T + 3531 T^{2} - 70 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 80 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 56 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 22 T - 3237 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 110 T + 6771 T^{2} + 110 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 160 T + 17679 T^{2} + 160 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 130 T + p^{2} T^{2} )^{2} \) |
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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
−11.48950669104572476838757929399, −11.42764764534448866915878488045, −10.86483864038721866654262165860, −10.45500193504466573583405051820, −9.786939398640770324681309439557, −9.684102258089983676743458883068, −8.687848399865716268265910169773, −8.324715172802997450492952979663, −7.46367641412922211947498409705, −7.03510096798253160591415062230, −6.40029985235839871257634173868, −6.21643905465330138286464466350, −5.53085309479530702136469362043, −5.35944364862405960347578800248, −4.55313083814263969586736586622, −4.07502475610214170738817076965, −3.30727797956611518042803844023, −2.70658746870262121974310267340, −1.90310670384207047914083062323, −1.54425538553117347082213142157,
1.54425538553117347082213142157, 1.90310670384207047914083062323, 2.70658746870262121974310267340, 3.30727797956611518042803844023, 4.07502475610214170738817076965, 4.55313083814263969586736586622, 5.35944364862405960347578800248, 5.53085309479530702136469362043, 6.21643905465330138286464466350, 6.40029985235839871257634173868, 7.03510096798253160591415062230, 7.46367641412922211947498409705, 8.324715172802997450492952979663, 8.687848399865716268265910169773, 9.684102258089983676743458883068, 9.786939398640770324681309439557, 10.45500193504466573583405051820, 10.86483864038721866654262165860, 11.42764764534448866915878488045, 11.48950669104572476838757929399
