| L(s) = 1 | − 2-s + 5·7-s + 8-s − 12·11-s − 5·13-s − 5·14-s − 16-s + 6·17-s + 4·19-s + 12·22-s − 12·23-s − 10·25-s + 5·26-s + 6·29-s + 31-s − 6·34-s + 37-s − 4·38-s − 6·41-s + 43-s + 12·46-s − 6·47-s + 18·49-s + 10·50-s − 6·53-s + 5·56-s − 6·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.88·7-s + 0.353·8-s − 3.61·11-s − 1.38·13-s − 1.33·14-s − 1/4·16-s + 1.45·17-s + 0.917·19-s + 2.55·22-s − 2.50·23-s − 2·25-s + 0.980·26-s + 1.11·29-s + 0.179·31-s − 1.02·34-s + 0.164·37-s − 0.648·38-s − 0.937·41-s + 0.152·43-s + 1.76·46-s − 0.875·47-s + 18/7·49-s + 1.41·50-s − 0.824·53-s + 0.668·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4268110236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4268110236\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - T - 66 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−9.958551526223595855206332710371, −9.800126945367223941535730406455, −9.460546799018121678264758689516, −8.302615108749417880550665931833, −8.242833893349697339424528780741, −8.008131415707988281563912985988, −7.83662499241586706486499338570, −7.40282711257185787720197550572, −7.13317487854057583238249128023, −5.88254054680190165166433383901, −5.76931195722910655549769074273, −5.16058731088957680612920971133, −5.12998615067236608923242002615, −4.48867585967093749663461055931, −4.07328144093397480858179774440, −2.87934063903172108645609724631, −2.84055454104471899862961679408, −1.93404512240286852963362138812, −1.68819024478826595079496178551, −0.30682151304072432538144532853,
0.30682151304072432538144532853, 1.68819024478826595079496178551, 1.93404512240286852963362138812, 2.84055454104471899862961679408, 2.87934063903172108645609724631, 4.07328144093397480858179774440, 4.48867585967093749663461055931, 5.12998615067236608923242002615, 5.16058731088957680612920971133, 5.76931195722910655549769074273, 5.88254054680190165166433383901, 7.13317487854057583238249128023, 7.40282711257185787720197550572, 7.83662499241586706486499338570, 8.008131415707988281563912985988, 8.242833893349697339424528780741, 8.302615108749417880550665931833, 9.460546799018121678264758689516, 9.800126945367223941535730406455, 9.958551526223595855206332710371
