| L(s) = 1 | − 4·3-s + 13·4-s + 58·9-s − 52·12-s + 64·16-s + 234·17-s − 156·23-s − 494·25-s − 416·27-s + 282·29-s + 754·36-s + 208·43-s − 256·48-s + 494·49-s − 936·51-s + 372·53-s − 290·61-s + 299·64-s + 3.04e3·68-s + 624·69-s + 1.97e3·75-s + 5.10e3·79-s + 1.96e3·81-s − 1.12e3·87-s − 2.02e3·92-s − 6.42e3·100-s + 858·101-s + ⋯ |
| L(s) = 1 | − 0.769·3-s + 13/8·4-s + 2.14·9-s − 1.25·12-s + 16-s + 3.33·17-s − 1.41·23-s − 3.95·25-s − 2.96·27-s + 1.80·29-s + 3.49·36-s + 0.737·43-s − 0.769·48-s + 1.44·49-s − 2.56·51-s + 0.964·53-s − 0.608·61-s + 0.583·64-s + 5.42·68-s + 1.08·69-s + 3.04·75-s + 7.26·79-s + 2.68·81-s − 1.39·87-s − 2.29·92-s − 6.42·100-s + 0.845·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.565403129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.565403129\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 13 T^{2} + 105 T^{4} - 13 p^{6} T^{6} + p^{12} T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + 2 T - 23 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 247 T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - 494 T^{2} + 126387 T^{4} - 494 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 2470 T^{2} + 4329339 T^{4} - 2470 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 117 T + 8776 T^{2} - 117 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 650 T^{2} - 46623381 T^{4} - 650 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + 78 T - 6083 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 141 T - 4508 T^{2} - 141 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 35282 T^{2} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 - 80639 T^{2} + 3936921912 T^{4} - 80639 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^3$ | \( 1 - 63895 T^{2} - 667533216 T^{4} - 63895 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 104 T - 68691 T^{2} - 104 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 116818 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 93 T + p^{3} T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - 330070 T^{2} + 66765671259 T^{4} - 330070 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 + 145 T - 205956 T^{2} + 145 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 16822 T^{2} - 90175402485 T^{4} + 16822 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 400478 T^{2} + 32282344563 T^{4} + 400478 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2$ | \( ( 1 + 567359 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 1276 T + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 519766 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^3$ | \( 1 - 455650 T^{2} - 289364368461 T^{4} - 455650 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^3$ | \( 1 - 1784978 T^{2} + 2353174455555 T^{4} - 1784978 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.132368863614644583100601364512, −8.452135781620371977430298683931, −8.004059712867094393146556608061, −7.76964545768307212282114741115, −7.68816290506712583437886042950, −7.56405279364463047215064457009, −7.50495917969218783065890384682, −6.75006598656477399595559965963, −6.60603704777648369113821010306, −6.42747578087283493493011287783, −5.91070358036158874848223804640, −5.87004057268646170482864223630, −5.44271749200565553006714038086, −5.40235960257539728770678959734, −4.75840455120789094673337462944, −4.33967047057970719492919851472, −3.84851002555649470570630737289, −3.77035506015757454564639316973, −3.52566074895650894001091441065, −2.86185262921755595038003357780, −2.15043839107452258692759141424, −1.94098282975302134432456121396, −1.77478074190564237779904645005, −0.890123206138796746847547626870, −0.68208662283862669500559305946,
0.68208662283862669500559305946, 0.890123206138796746847547626870, 1.77478074190564237779904645005, 1.94098282975302134432456121396, 2.15043839107452258692759141424, 2.86185262921755595038003357780, 3.52566074895650894001091441065, 3.77035506015757454564639316973, 3.84851002555649470570630737289, 4.33967047057970719492919851472, 4.75840455120789094673337462944, 5.40235960257539728770678959734, 5.44271749200565553006714038086, 5.87004057268646170482864223630, 5.91070358036158874848223804640, 6.42747578087283493493011287783, 6.60603704777648369113821010306, 6.75006598656477399595559965963, 7.50495917969218783065890384682, 7.56405279364463047215064457009, 7.68816290506712583437886042950, 7.76964545768307212282114741115, 8.004059712867094393146556608061, 8.452135781620371977430298683931, 9.132368863614644583100601364512
