| L(s) = 1 | + 6·2-s + 2·3-s + 24·4-s + 20·5-s + 12·6-s + 24·7-s + 80·8-s − 36·9-s + 120·10-s + 10·11-s + 48·12-s − 4·13-s + 144·14-s + 40·15-s + 240·16-s − 66·17-s − 216·18-s − 164·19-s + 480·20-s + 48·21-s + 60·22-s − 204·23-s + 160·24-s + 52·25-s − 24·26-s − 140·27-s + 576·28-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 0.384·3-s + 3·4-s + 1.78·5-s + 0.816·6-s + 1.29·7-s + 3.53·8-s − 4/3·9-s + 3.79·10-s + 0.274·11-s + 1.15·12-s − 0.0853·13-s + 2.74·14-s + 0.688·15-s + 15/4·16-s − 0.941·17-s − 2.82·18-s − 1.98·19-s + 5.36·20-s + 0.498·21-s + 0.581·22-s − 1.84·23-s + 1.36·24-s + 0.415·25-s − 0.181·26-s − 0.997·27-s + 3.88·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195112 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195112 ^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(11.66269654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.66269654\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 29 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 - 2 T + 40 T^{2} - 4 p T^{3} + 40 p^{3} T^{4} - 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 4 p T + 348 T^{2} - 4838 T^{3} + 348 p^{3} T^{4} - 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 24 T + 533 T^{2} - 1584 p T^{3} + 533 p^{3} T^{4} - 24 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 10 T + 3760 T^{2} - 24196 T^{3} + 3760 p^{3} T^{4} - 10 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 4 T + 972 T^{2} + 149282 T^{3} + 972 p^{3} T^{4} + 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 66 T + 4079 T^{2} - 30852 T^{3} + 4079 p^{3} T^{4} + 66 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 164 T + 20453 T^{2} + 1585304 T^{3} + 20453 p^{3} T^{4} + 164 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 204 T + 40785 T^{2} + 4286760 T^{3} + 40785 p^{3} T^{4} + 204 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 86 T + 41284 T^{2} + 357880 T^{3} + 41284 p^{3} T^{4} + 86 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 42 T + 72503 T^{2} - 3430044 T^{3} + 72503 p^{3} T^{4} + 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 562 T + 309195 T^{2} - 83449252 T^{3} + 309195 p^{3} T^{4} - 562 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 18 T + 228152 T^{2} - 2665764 T^{3} + 228152 p^{3} T^{4} - 18 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 654 T + 409484 T^{2} - 139225608 T^{3} + 409484 p^{3} T^{4} - 654 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 712 T + 96060 T^{2} + 40119698 T^{3} + 96060 p^{3} T^{4} - 712 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 184 T + 304693 T^{2} - 18216544 T^{3} + 304693 p^{3} T^{4} - 184 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 322 T + 292571 T^{2} - 151430188 T^{3} + 292571 p^{3} T^{4} - 322 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 228 T + 574449 T^{2} + 184756120 T^{3} + 574449 p^{3} T^{4} + 228 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 52 T + 1073569 T^{2} + 37222072 T^{3} + 1073569 p^{3} T^{4} + 52 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 494 T + 1173163 T^{2} + 374938588 T^{3} + 1173163 p^{3} T^{4} + 494 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 2110 T + 2879660 T^{2} + 2365811752 T^{3} + 2879660 p^{3} T^{4} + 2110 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 288 T + 676365 T^{2} - 108256752 T^{3} + 676365 p^{3} T^{4} + 288 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 914 T + 1450107 T^{2} - 690671780 T^{3} + 1450107 p^{3} T^{4} - 914 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 218 T + 1237251 T^{2} - 414931540 T^{3} + 1237251 p^{3} T^{4} - 218 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.44597441941472705350980693942, −13.00763983654731322248919301091, −12.57758249238211522831394727583, −12.15337541321050127422455475899, −11.67666899847135247567434140918, −11.36819564812106739379756867763, −11.12319808308446208598881573956, −10.53714188433198907932876274037, −10.19856902046352002851592275966, −9.868617660089261064159299073627, −8.914897700803803362109616469528, −8.760678491850353688027892860468, −8.364916456244066399396894411729, −7.56125960176114894214730510772, −7.31786814967812998853875691882, −6.39247721929765490332409662893, −6.02323675885339702628982660193, −5.83947047841683236437293032818, −5.53481451521017176833109710117, −4.67410619931427632141513525141, −4.26843161513974804705473548374, −3.81406849229762969633251061186, −2.55357104464373964279627950597, −2.25898666150820434096587685930, −1.88016279849672962719054285816,
1.88016279849672962719054285816, 2.25898666150820434096587685930, 2.55357104464373964279627950597, 3.81406849229762969633251061186, 4.26843161513974804705473548374, 4.67410619931427632141513525141, 5.53481451521017176833109710117, 5.83947047841683236437293032818, 6.02323675885339702628982660193, 6.39247721929765490332409662893, 7.31786814967812998853875691882, 7.56125960176114894214730510772, 8.364916456244066399396894411729, 8.760678491850353688027892860468, 8.914897700803803362109616469528, 9.868617660089261064159299073627, 10.19856902046352002851592275966, 10.53714188433198907932876274037, 11.12319808308446208598881573956, 11.36819564812106739379756867763, 11.67666899847135247567434140918, 12.15337541321050127422455475899, 12.57758249238211522831394727583, 13.00763983654731322248919301091, 13.44597441941472705350980693942
