| L(s) = 1 | + 2-s − 3-s − 6-s + 7-s − 2·11-s − 2·13-s + 14-s + 19-s − 21-s − 2·22-s − 23-s + 2·25-s − 2·26-s − 32-s + 2·33-s + 38-s + 2·39-s + 41-s − 42-s − 46-s + 2·50-s − 53-s − 57-s − 64-s + 2·66-s + 69-s + 73-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s − 6-s + 7-s − 2·11-s − 2·13-s + 14-s + 19-s − 21-s − 2·22-s − 23-s + 2·25-s − 2·26-s − 32-s + 2·33-s + 38-s + 2·39-s + 41-s − 42-s − 46-s + 2·50-s − 53-s − 57-s − 64-s + 2·66-s + 69-s + 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20449 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4370540869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4370540869\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 7 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 23 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−13.60486632803815923946613303662, −13.01428957171829116532910775374, −12.49653956081239081415209781238, −12.36739753161912299402971421224, −11.67030199132299516852202375700, −11.09964703010420586370456732131, −10.82066324397992018341842346619, −10.21318557697500063287799539775, −9.724324737236978039854594585250, −9.004257973228336367496572581079, −8.129979544218070959974762466048, −7.66026602909536586139046396552, −7.34040097683464702793451134111, −6.44399786173049336138211999786, −5.40644236408523558557252902644, −5.31421961759558757154414127899, −4.86221255192137125487916254236, −4.38802451253424855324823205856, −3.07141793703332287184327843999, −2.27203371878102644767358401045,
2.27203371878102644767358401045, 3.07141793703332287184327843999, 4.38802451253424855324823205856, 4.86221255192137125487916254236, 5.31421961759558757154414127899, 5.40644236408523558557252902644, 6.44399786173049336138211999786, 7.34040097683464702793451134111, 7.66026602909536586139046396552, 8.129979544218070959974762466048, 9.004257973228336367496572581079, 9.724324737236978039854594585250, 10.21318557697500063287799539775, 10.82066324397992018341842346619, 11.09964703010420586370456732131, 11.67030199132299516852202375700, 12.36739753161912299402971421224, 12.49653956081239081415209781238, 13.01428957171829116532910775374, 13.60486632803815923946613303662
