| L(s) = 1 | − 8·4-s − 54·5-s − 28·7-s + 54·11-s − 918·17-s + 30·19-s + 432·20-s + 486·23-s + 547·25-s + 224·28-s − 3.24e3·29-s − 546·31-s + 1.51e3·35-s − 446·37-s + 2.34e3·43-s − 432·44-s − 702·47-s − 4.21e3·49-s − 2.75e3·53-s − 2.91e3·55-s − 1.23e4·59-s + 7.68e3·61-s + 512·64-s − 5.06e3·67-s + 7.34e3·68-s − 1.87e4·71-s − 1.72e4·73-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 2.15·5-s − 4/7·7-s + 0.446·11-s − 3.17·17-s + 0.0831·19-s + 1.07·20-s + 0.918·23-s + 0.875·25-s + 2/7·28-s − 3.85·29-s − 0.568·31-s + 1.23·35-s − 0.325·37-s + 1.26·43-s − 0.223·44-s − 0.317·47-s − 1.75·49-s − 0.980·53-s − 0.963·55-s − 3.55·59-s + 2.06·61-s + 1/8·64-s − 1.12·67-s + 1.58·68-s − 3.72·71-s − 3.24·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1008945256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1008945256\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + 2 p T + p^{4} T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 54 T + 2369 T^{2} + 75438 T^{3} + 2168484 T^{4} + 75438 p^{4} T^{5} + 2369 p^{8} T^{6} + 54 p^{12} T^{7} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 54 T - 2437 p T^{2} - 23814 T^{3} + 626404692 T^{4} - 23814 p^{4} T^{5} - 2437 p^{9} T^{6} - 54 p^{12} T^{7} + p^{16} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 51652 T^{2} + 2274555846 T^{4} - 51652 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 54 p T + 493601 T^{2} + 11485422 p T^{3} + 61724271876 T^{4} + 11485422 p^{5} T^{5} + 493601 p^{8} T^{6} + 54 p^{13} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 30 T + 66617 T^{2} - 1989510 T^{3} - 12546522252 T^{4} - 1989510 p^{4} T^{5} + 66617 p^{8} T^{6} - 30 p^{12} T^{7} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 486 T + 78265 T^{2} + 195250986 T^{3} - 119466109356 T^{4} + 195250986 p^{4} T^{5} + 78265 p^{8} T^{6} - 486 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 1620 T + 2066054 T^{2} + 1620 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 546 T + 1193657 T^{2} + 597479610 T^{3} + 436340752596 T^{4} + 597479610 p^{4} T^{5} + 1193657 p^{8} T^{6} + 546 p^{12} T^{7} + p^{16} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 446 T - 3015935 T^{2} - 237928066 T^{3} + 6449986888804 T^{4} - 237928066 p^{4} T^{5} - 3015935 p^{8} T^{6} + 446 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 9195268 T^{2} + 37043496050310 T^{4} - 9195268 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 1172 T + 3821766 T^{2} - 1172 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 702 T + 7568153 T^{2} + 5197527270 T^{3} + 31807801869972 T^{4} + 5197527270 p^{4} T^{5} + 7568153 p^{8} T^{6} + 702 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 2754 T - 5299967 T^{2} - 7976903166 T^{3} + 43904732373732 T^{4} - 7976903166 p^{4} T^{5} - 5299967 p^{8} T^{6} + 2754 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 12366 T + 87438953 T^{2} + 450942278166 T^{3} + 1800614696429652 T^{4} + 450942278166 p^{4} T^{5} + 87438953 p^{8} T^{6} + 12366 p^{12} T^{7} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 126 p T + 39234641 T^{2} - 2462431734 p T^{3} + 462871617507012 T^{4} - 2462431734 p^{5} T^{5} + 39234641 p^{8} T^{6} - 126 p^{13} T^{7} + p^{16} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 5062 T + 1333849 T^{2} - 81053994314 T^{3} - 332413001385740 T^{4} - 81053994314 p^{4} T^{5} + 1333849 p^{8} T^{6} + 5062 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 9396 T + 71231078 T^{2} + 9396 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 17274 T + 161686097 T^{2} + 1074829823970 T^{3} + 5889761488255716 T^{4} + 1074829823970 p^{4} T^{5} + 161686097 p^{8} T^{6} + 17274 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 794 T - 69869063 T^{2} + 5876126422 T^{3} + 3428515016079124 T^{4} + 5876126422 p^{4} T^{5} - 69869063 p^{8} T^{6} - 794 p^{12} T^{7} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 153397060 T^{2} + 10369989980918982 T^{4} - 153397060 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12474 T + 182683793 T^{2} - 1631810023074 T^{3} + 16430717819326692 T^{4} - 1631810023074 p^{4} T^{5} + 182683793 p^{8} T^{6} - 12474 p^{12} T^{7} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 281924740 T^{2} + 35525126061727494 T^{4} - 281924740 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−8.967675004036889008386826726971, −8.866249361561167734611652187958, −8.819917755621181532938855623918, −8.318732417181404635030459666059, −7.65370036214452377938444927827, −7.59618772528969111870370187584, −7.56965561997432072848876709785, −7.23627544153522515157731880722, −6.78876837568254170423416338265, −6.62875913729245802113002275962, −5.96161419763693693373347792133, −5.79460759568193234974463293120, −5.76675728676495871922716618321, −4.68676026331060835200944169048, −4.63052295457853540552753112755, −4.44533767684630606836265711957, −4.23510753861796667746183380517, −3.50053039792297461640163640455, −3.48126444802993689901658523548, −3.21058222920407238829783308329, −2.42195404849023254016549741432, −1.74031593869251843742930281111, −1.67681820752629505081169034069, −0.34575469021196868838455809089, −0.15858975285573933296873637890,
0.15858975285573933296873637890, 0.34575469021196868838455809089, 1.67681820752629505081169034069, 1.74031593869251843742930281111, 2.42195404849023254016549741432, 3.21058222920407238829783308329, 3.48126444802993689901658523548, 3.50053039792297461640163640455, 4.23510753861796667746183380517, 4.44533767684630606836265711957, 4.63052295457853540552753112755, 4.68676026331060835200944169048, 5.76675728676495871922716618321, 5.79460759568193234974463293120, 5.96161419763693693373347792133, 6.62875913729245802113002275962, 6.78876837568254170423416338265, 7.23627544153522515157731880722, 7.56965561997432072848876709785, 7.59618772528969111870370187584, 7.65370036214452377938444927827, 8.318732417181404635030459666059, 8.819917755621181532938855623918, 8.866249361561167734611652187958, 8.967675004036889008386826726971
