| L(s) = 1 | + 6·2-s − 5·3-s + 24·4-s + 5-s − 30·6-s + 26·7-s + 80·8-s − 9-s + 6·10-s + 4·11-s − 120·12-s + 129·13-s + 156·14-s − 5·15-s + 240·16-s + 51·17-s − 6·18-s + 24·20-s − 130·21-s + 24·22-s − 47·23-s − 400·24-s − 18·25-s + 774·26-s − 28·27-s + 624·28-s − 125·29-s + ⋯ |
| L(s) = 1 | + 2.12·2-s − 0.962·3-s + 3·4-s + 0.0894·5-s − 2.04·6-s + 1.40·7-s + 3.53·8-s − 0.0370·9-s + 0.189·10-s + 0.109·11-s − 2.88·12-s + 2.75·13-s + 2.97·14-s − 0.0860·15-s + 15/4·16-s + 0.727·17-s − 0.0785·18-s + 0.268·20-s − 1.35·21-s + 0.232·22-s − 0.426·23-s − 3.40·24-s − 0.143·25-s + 5.83·26-s − 0.199·27-s + 4.21·28-s − 0.800·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(24.13290820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(24.13290820\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{3} \) |
| 19 | | \( 1 \) |
| good | 3 | $S_4\times C_2$ | \( 1 + 5 T + 26 T^{2} + 163 T^{3} + 26 p^{3} T^{4} + 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 5 | $S_4\times C_2$ | \( 1 - T + 19 T^{2} + 1688 T^{3} + 19 p^{3} T^{4} - p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 - 26 T + 1191 T^{2} - 17932 T^{3} + 1191 p^{3} T^{4} - 26 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 62 p T^{2} + 39332 T^{3} + 62 p^{4} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 129 T + 10575 T^{2} - 555454 T^{3} + 10575 p^{3} T^{4} - 129 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 3 p T + 12507 T^{2} - 394854 T^{3} + 12507 p^{3} T^{4} - 3 p^{7} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 + 47 T + 35053 T^{2} + 1076204 T^{3} + 35053 p^{3} T^{4} + 47 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 125 T + 23767 T^{2} + 8534000 T^{3} + 23767 p^{3} T^{4} + 125 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 50 T + 37223 T^{2} - 6788948 T^{3} + 37223 p^{3} T^{4} - 50 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 188 T + 151301 T^{2} - 18957524 T^{3} + 151301 p^{3} T^{4} - 188 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 475 T + 225622 T^{2} - 56464759 T^{3} + 225622 p^{3} T^{4} - 475 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 73 T + 126797 T^{2} + 4494434 T^{3} + 126797 p^{3} T^{4} - 73 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 241 T + 294481 T^{2} - 44975956 T^{3} + 294481 p^{3} T^{4} - 241 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 29 T + 407983 T^{2} - 5339846 T^{3} + 407983 p^{3} T^{4} - 29 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 1065 T + 972522 T^{2} + 473334231 T^{3} + 972522 p^{3} T^{4} + 1065 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 981 T + 949611 T^{2} - 464210144 T^{3} + 949611 p^{3} T^{4} - 981 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 877 T + 841242 T^{2} - 528520259 T^{3} + 841242 p^{3} T^{4} - 877 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 2135 T + 2443777 T^{2} - 354926 p^{2} T^{3} + 2443777 p^{3} T^{4} - 2135 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 667 T + 1261314 T^{2} + 519793475 T^{3} + 1261314 p^{3} T^{4} + 667 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 1671 T + 1726197 T^{2} - 1253980738 T^{3} + 1726197 p^{3} T^{4} - 1671 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 588 T + 867318 T^{2} - 834896496 T^{3} + 867318 p^{3} T^{4} - 588 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 693 T + 1798515 T^{2} - 924765786 T^{3} + 1798515 p^{3} T^{4} - 693 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 985 T + 305282 T^{2} - 810153335 T^{3} + 305282 p^{3} T^{4} + 985 p^{6} T^{5} + p^{9} T^{6} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.784479678732299111053414420277, −8.223657888925175521195428947353, −8.138901590661601435787105360696, −8.072992311678736786135610589513, −7.56854499095053166933907900294, −7.46213092620544637845981847959, −6.74283023280756988028876696629, −6.38068595481065757137992084559, −6.34555315041668185002228221819, −6.25354325146888671986672824753, −5.56419795430675214863241506179, −5.49082523962659954227784121273, −5.35472624509981091232202125152, −4.97284634616562419465005000310, −4.49602229177165911550055155313, −4.21706076352275178144575301646, −3.81068834939158711288749782966, −3.59713694914728375355793672053, −3.46345836419325938700215050454, −2.59954485328303747879925136529, −2.43257057866834639817571107804, −1.82551620630567192505185668437, −1.38765820362110525043409382971, −1.11197933059196863024572438828, −0.58672617477165295743217855787,
0.58672617477165295743217855787, 1.11197933059196863024572438828, 1.38765820362110525043409382971, 1.82551620630567192505185668437, 2.43257057866834639817571107804, 2.59954485328303747879925136529, 3.46345836419325938700215050454, 3.59713694914728375355793672053, 3.81068834939158711288749782966, 4.21706076352275178144575301646, 4.49602229177165911550055155313, 4.97284634616562419465005000310, 5.35472624509981091232202125152, 5.49082523962659954227784121273, 5.56419795430675214863241506179, 6.25354325146888671986672824753, 6.34555315041668185002228221819, 6.38068595481065757137992084559, 6.74283023280756988028876696629, 7.46213092620544637845981847959, 7.56854499095053166933907900294, 8.072992311678736786135610589513, 8.138901590661601435787105360696, 8.223657888925175521195428947353, 8.784479678732299111053414420277
