| L(s) = 1 | + 4-s + 9-s + 10·11-s − 4·19-s + 10·29-s − 44·31-s + 36-s + 4·41-s + 10·44-s − 10·49-s − 30·59-s − 20·61-s − 64-s − 4·76-s + 44·79-s + 4·89-s + 10·99-s − 4·101-s + 8·109-s + 10·116-s + 47·121-s − 44·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s + 3.01·11-s − 0.917·19-s + 1.85·29-s − 7.90·31-s + 1/6·36-s + 0.624·41-s + 1.50·44-s − 1.42·49-s − 3.90·59-s − 2.56·61-s − 1/8·64-s − 0.458·76-s + 4.95·79-s + 0.423·89-s + 1.00·99-s − 0.398·101-s + 0.766·109-s + 0.928·116-s + 4.27·121-s − 3.95·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.577369893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.577369893\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| good | 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 8 T + 47 T^{2} - 8 p T^{3} + p^{2} T^{4} )( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 - 35 T^{2} - 624 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 2 T - 85 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.38983457656877956405881938589, −6.32800109421215922739135478849, −6.18384023069064505850465277195, −5.95518851020721700730465212473, −5.94449468473898000115201957124, −5.44100252246311032470146464377, −5.28457297368841068058511503464, −4.99424847125880081981777695669, −4.86609438457879286862394700778, −4.46293730273541636729741416599, −4.42898146892428771487688022695, −4.03454564218917444515837502544, −3.83444926623896607947559928947, −3.72492783203526042318530085444, −3.44365348809485501723750755906, −3.26240465982343213698943656388, −3.11107804089958557859208640472, −2.70326902081056409607779985780, −2.00283536386392535551068372740, −1.98340693993395811746211612441, −1.87958672546825060921466633712, −1.51260533180443386977805239735, −1.42570614934174808288392368187, −0.812420861447181400665992279812, −0.20962138138586612378228140035,
0.20962138138586612378228140035, 0.812420861447181400665992279812, 1.42570614934174808288392368187, 1.51260533180443386977805239735, 1.87958672546825060921466633712, 1.98340693993395811746211612441, 2.00283536386392535551068372740, 2.70326902081056409607779985780, 3.11107804089958557859208640472, 3.26240465982343213698943656388, 3.44365348809485501723750755906, 3.72492783203526042318530085444, 3.83444926623896607947559928947, 4.03454564218917444515837502544, 4.42898146892428771487688022695, 4.46293730273541636729741416599, 4.86609438457879286862394700778, 4.99424847125880081981777695669, 5.28457297368841068058511503464, 5.44100252246311032470146464377, 5.94449468473898000115201957124, 5.95518851020721700730465212473, 6.18384023069064505850465277195, 6.32800109421215922739135478849, 6.38983457656877956405881938589
