| L(s) = 1 | − 48·2-s + 1.53e3·4-s + 248·5-s − 2.95e3·7-s − 4.09e4·8-s − 3.99e4·9-s − 1.19e4·10-s − 3.13e4·11-s + 8.56e4·13-s + 1.41e5·14-s + 9.83e5·16-s + 9.05e5·17-s + 1.91e6·18-s + 1.72e6·19-s + 3.80e5·20-s + 1.50e6·22-s + 2.13e6·23-s − 8.61e5·25-s − 4.11e6·26-s + 1.01e6·27-s − 4.54e6·28-s + 3.72e5·29-s − 1.44e6·31-s − 2.20e7·32-s − 4.34e7·34-s − 7.33e5·35-s − 6.13e7·36-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 3·4-s + 0.177·5-s − 0.465·7-s − 3.53·8-s − 2.02·9-s − 0.376·10-s − 0.645·11-s + 0.832·13-s + 0.987·14-s + 15/4·16-s + 2.62·17-s + 4.30·18-s + 3.03·19-s + 0.532·20-s + 1.36·22-s + 1.59·23-s − 0.441·25-s − 1.76·26-s + 0.366·27-s − 1.39·28-s + 0.0977·29-s − 0.280·31-s − 3.71·32-s − 5.57·34-s − 0.0825·35-s − 6.08·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17576 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17576 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.196526134\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.196526134\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_1$ | \( ( 1 + p^{4} T )^{3} \) |
| 13 | $C_1$ | \( ( 1 - p^{4} T )^{3} \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 39956 T^{2} - 337636 p T^{3} + 39956 p^{9} T^{4} + p^{27} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - 248 T + 923292 T^{2} + 601794146 p T^{3} + 923292 p^{9} T^{4} - 248 p^{18} T^{5} + p^{27} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2956 T + 6282440 p T^{2} + 7372186912 p^{2} T^{3} + 6282440 p^{10} T^{4} + 2956 p^{18} T^{5} + p^{27} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 31324 T + 3108381357 T^{2} + 1295264796184 T^{3} + 3108381357 p^{9} T^{4} + 31324 p^{18} T^{5} + p^{27} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 905228 T + 543223862736 T^{2} - 225327251204805818 T^{3} + 543223862736 p^{9} T^{4} - 905228 p^{18} T^{5} + p^{27} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 1726316 T + 1900897565189 T^{2} - 1273642462589358776 T^{3} + 1900897565189 p^{9} T^{4} - 1726316 p^{18} T^{5} + p^{27} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 2135256 T - 1254922557 p T^{2} + 3475027705569520944 T^{3} - 1254922557 p^{10} T^{4} - 2135256 p^{18} T^{5} + p^{27} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 372426 T + 39292154724291 T^{2} - 12093573529001771772 T^{3} + 39292154724291 p^{9} T^{4} - 372426 p^{18} T^{5} + p^{27} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 1444776 T + 20125130602413 T^{2} - 3600588039795005360 p T^{3} + 20125130602413 p^{9} T^{4} + 1444776 p^{18} T^{5} + p^{27} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 2809672 T + 118704084320492 T^{2} - \)\(91\!\cdots\!70\)\( T^{3} + 118704084320492 p^{9} T^{4} + 2809672 p^{18} T^{5} + p^{27} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 7424806 T + 385002837668763 T^{2} - \)\(24\!\cdots\!68\)\( T^{3} + 385002837668763 p^{9} T^{4} + 7424806 p^{18} T^{5} + p^{27} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 36070040 T + 1498362255411236 T^{2} - \)\(29\!\cdots\!92\)\( T^{3} + 1498362255411236 p^{9} T^{4} - 36070040 p^{18} T^{5} + p^{27} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 7032884 T + 1888660171993728 T^{2} - \)\(16\!\cdots\!08\)\( T^{3} + 1888660171993728 p^{9} T^{4} - 7032884 p^{18} T^{5} + p^{27} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 15011674 T + 8016410991974559 T^{2} + \)\(10\!\cdots\!84\)\( T^{3} + 8016410991974559 p^{9} T^{4} + 15011674 p^{18} T^{5} + p^{27} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 40286396 T + 2634081870833661 T^{2} + \)\(57\!\cdots\!12\)\( T^{3} + 2634081870833661 p^{9} T^{4} + 40286396 p^{18} T^{5} + p^{27} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 230710478 T + 49861187231005559 T^{2} - \)\(56\!\cdots\!04\)\( T^{3} + 49861187231005559 p^{9} T^{4} - 230710478 p^{18} T^{5} + p^{27} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 201453084 T + 92950677525555765 T^{2} - \)\(11\!\cdots\!12\)\( T^{3} + 92950677525555765 p^{9} T^{4} - 201453084 p^{18} T^{5} + p^{27} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 99165572 T + 47050108897302528 T^{2} + \)\(11\!\cdots\!48\)\( T^{3} + 47050108897302528 p^{9} T^{4} + 99165572 p^{18} T^{5} + p^{27} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 350942946 T + 156947510226839943 T^{2} + \)\(36\!\cdots\!96\)\( T^{3} + 156947510226839943 p^{9} T^{4} + 350942946 p^{18} T^{5} + p^{27} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 138794592 T + 177109833453405357 T^{2} + \)\(67\!\cdots\!80\)\( T^{3} + 177109833453405357 p^{9} T^{4} - 138794592 p^{18} T^{5} + p^{27} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 1364469516 T + 961277392341181833 T^{2} + \)\(46\!\cdots\!76\)\( T^{3} + 961277392341181833 p^{9} T^{4} + 1364469516 p^{18} T^{5} + p^{27} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 709437058 T + 1051916775683610615 T^{2} + \)\(49\!\cdots\!00\)\( T^{3} + 1051916775683610615 p^{9} T^{4} + 709437058 p^{18} T^{5} + p^{27} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 362189698 T + 980396350748438111 T^{2} + \)\(23\!\cdots\!32\)\( T^{3} + 980396350748438111 p^{9} T^{4} + 362189698 p^{18} T^{5} + p^{27} T^{6} \) |
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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
−14.08675914348512984829736952641, −13.17800317553637696510245991683, −12.63020318103905995050823443547, −12.15838948731671798459949315541, −11.56169234923557941638289628921, −11.36453446435980598885757793076, −11.12931914039912323815731995571, −10.30389553703651258467397088437, −9.875005297074567728375589192647, −9.763188566684373617740618452605, −9.039484673603341694340625655664, −8.726340941141516650853413775151, −8.252560588214758011598271335686, −7.55533746520910779436546792741, −7.54660710920621517861496736662, −6.78227204660263847239710173565, −5.88020474365813684677509836470, −5.47438258662496058177250840780, −5.42741914896907171850221760046, −3.33317306650557759935701915811, −3.12450037071288896300264883051, −2.79540920494524299639557192985, −1.53739968747364262683868535690, −0.844935264710193199222780306512, −0.60148432085787545058140478930,
0.60148432085787545058140478930, 0.844935264710193199222780306512, 1.53739968747364262683868535690, 2.79540920494524299639557192985, 3.12450037071288896300264883051, 3.33317306650557759935701915811, 5.42741914896907171850221760046, 5.47438258662496058177250840780, 5.88020474365813684677509836470, 6.78227204660263847239710173565, 7.54660710920621517861496736662, 7.55533746520910779436546792741, 8.252560588214758011598271335686, 8.726340941141516650853413775151, 9.039484673603341694340625655664, 9.763188566684373617740618452605, 9.875005297074567728375589192647, 10.30389553703651258467397088437, 11.12931914039912323815731995571, 11.36453446435980598885757793076, 11.56169234923557941638289628921, 12.15838948731671798459949315541, 12.63020318103905995050823443547, 13.17800317553637696510245991683, 14.08675914348512984829736952641
