| L(s) = 1 | + 8·2-s + 40·4-s + 160·8-s + 560·16-s + 110·17-s + 42·19-s + 122·23-s + 400·31-s + 1.79e3·32-s + 880·34-s + 336·38-s + 976·46-s + 772·47-s − 149·49-s + 372·53-s + 1.56e3·61-s + 3.20e3·62-s + 5.37e3·64-s + 4.40e3·68-s + 1.68e3·76-s + 3.25e3·79-s − 1.26e3·83-s + 4.88e3·92-s + 6.17e3·94-s − 1.19e3·98-s + 2.97e3·106-s + 1.75e3·107-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s + 7.07·8-s + 35/4·16-s + 1.56·17-s + 0.507·19-s + 1.10·23-s + 2.31·31-s + 9.89·32-s + 4.43·34-s + 1.43·38-s + 3.12·46-s + 2.39·47-s − 0.434·49-s + 0.964·53-s + 3.27·61-s + 6.55·62-s + 21/2·64-s + 7.84·68-s + 2.53·76-s + 4.63·79-s − 1.67·83-s + 5.53·92-s + 6.77·94-s − 1.22·98-s + 2.72·106-s + 1.58·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(136.3409414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(136.3409414\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2 \wr C_2$ | \( 1 + 149 T^{2} + 233592 T^{4} + 149 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 4101 T^{2} + 7740416 T^{4} + 4101 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 2120 T^{2} + 1373118 T^{4} - 2120 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 55 T + 3326 T^{2} - 55 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 21 T + 6572 T^{2} - 21 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $D_{4}$ | \( ( 1 - 61 T + 18008 T^{2} - 61 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2 \wr C_2$ | \( 1 - 616 T^{2} + 428009406 T^{4} - 616 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 200 T + 40557 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 + 69800 T^{2} + 2202254718 T^{4} + 69800 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 48252 T^{2} + 6320632358 T^{4} + 48252 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $C_2^2 \wr C_2$ | \( 1 + 205028 T^{2} + 20876286294 T^{4} + 205028 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 386 T + 215870 T^{2} - 386 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 186 T + 190303 T^{2} - 186 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 255108 T^{2} + 87834083798 T^{4} + 255108 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 781 T + 425046 T^{2} - 781 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 + 494624 T^{2} + 227448487182 T^{4} + 494624 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 + 1158944 T^{2} + 586110804126 T^{4} + 1158944 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 + 893621 T^{2} + 498962646732 T^{4} + 893621 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 1629 T + 1584182 T^{2} - 1629 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 632 T + 982205 T^{2} + 632 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 2703228 T^{2} + 2819260745318 T^{4} + 2703228 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 3093809 T^{2} + 3981693701472 T^{4} + 3093809 p^{6} T^{6} + p^{12} T^{8} \) |
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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
−6.49933093744297761387649801662, −5.90343850098953392975302488952, −5.89100385312683108355812691859, −5.80044346914239773290532548622, −5.70592585716636983436902709671, −5.19140156664054333444352244131, −5.00874863727949046312734023458, −5.00622304486401073319576947456, −4.84754248818098598151332602821, −4.39850369100562704233597100077, −4.15076028466203537763157411586, −4.02858603477882456455285012070, −3.85432755193853576200509110948, −3.39329646643364823049019474234, −3.28807262986171640021481936877, −3.10376347459520159073986909763, −2.92840808508567749792014516111, −2.48847483860294666461962707680, −2.27639885723009769633796365227, −2.06413077139433504356070774102, −1.86020211489471953660964184844, −1.14129522776615376375465860892, −1.01503851785736268319138085634, −0.71643924554214015992397368587, −0.64785979481639218931301143314,
0.64785979481639218931301143314, 0.71643924554214015992397368587, 1.01503851785736268319138085634, 1.14129522776615376375465860892, 1.86020211489471953660964184844, 2.06413077139433504356070774102, 2.27639885723009769633796365227, 2.48847483860294666461962707680, 2.92840808508567749792014516111, 3.10376347459520159073986909763, 3.28807262986171640021481936877, 3.39329646643364823049019474234, 3.85432755193853576200509110948, 4.02858603477882456455285012070, 4.15076028466203537763157411586, 4.39850369100562704233597100077, 4.84754248818098598151332602821, 5.00622304486401073319576947456, 5.00874863727949046312734023458, 5.19140156664054333444352244131, 5.70592585716636983436902709671, 5.80044346914239773290532548622, 5.89100385312683108355812691859, 5.90343850098953392975302488952, 6.49933093744297761387649801662
