| L(s) = 1 | + 3·2-s − 7·4-s − 15·5-s − 5·7-s − 29·8-s − 45·10-s + 4·11-s − 125·13-s − 15·14-s + 17·16-s + 119·17-s − 57·19-s + 105·20-s + 12·22-s + 129·23-s + 150·25-s − 375·26-s + 35·28-s − 37·29-s + 90·31-s + 43·32-s + 357·34-s + 75·35-s − 250·37-s − 171·38-s + 435·40-s + 94·41-s + ⋯ |
| L(s) = 1 | + 1.06·2-s − 7/8·4-s − 1.34·5-s − 0.269·7-s − 1.28·8-s − 1.42·10-s + 0.109·11-s − 2.66·13-s − 0.286·14-s + 0.265·16-s + 1.69·17-s − 0.688·19-s + 1.17·20-s + 0.116·22-s + 1.16·23-s + 6/5·25-s − 2.82·26-s + 0.236·28-s − 0.236·29-s + 0.521·31-s + 0.237·32-s + 1.80·34-s + 0.362·35-s − 1.11·37-s − 0.729·38-s + 1.71·40-s + 0.358·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{6} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| 19 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 2 | $S_4\times C_2$ | \( 1 - 3 T + p^{4} T^{2} - 5 p^{3} T^{3} + p^{7} T^{4} - 3 p^{6} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 5 T + 661 T^{2} + 5394 T^{3} + 661 p^{3} T^{4} + 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 4 T + 1277 T^{2} - 50040 T^{3} + 1277 p^{3} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 125 T + 10065 T^{2} + 539892 T^{3} + 10065 p^{3} T^{4} + 125 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 7 p T + 16419 T^{2} - 64754 p T^{3} + 16419 p^{3} T^{4} - 7 p^{7} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 129 T + 31597 T^{2} - 2866082 T^{3} + 31597 p^{3} T^{4} - 129 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 37 T + 40615 T^{2} - 532450 T^{3} + 40615 p^{3} T^{4} + 37 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 90 T + 27877 T^{2} - 3967452 T^{3} + 27877 p^{3} T^{4} - 90 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 250 T + 152177 T^{2} + 23803448 T^{3} + 152177 p^{3} T^{4} + 250 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 94 T + 173239 T^{2} - 11698564 T^{3} + 173239 p^{3} T^{4} - 94 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 210 T + 194253 T^{2} + 33060724 T^{3} + 194253 p^{3} T^{4} + 210 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 524 T + 255353 T^{2} + 83979368 T^{3} + 255353 p^{3} T^{4} + 524 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 1417 T + 1101521 T^{2} - 520917204 T^{3} + 1101521 p^{3} T^{4} - 1417 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 15 T + 189465 T^{2} + 1625338 T^{3} + 189465 p^{3} T^{4} + 15 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 912 T + 591919 T^{2} + 251223968 T^{3} + 591919 p^{3} T^{4} + 912 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 237 T + 199291 T^{2} - 319689904 T^{3} + 199291 p^{3} T^{4} - 237 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 124 T + 876893 T^{2} - 59492776 T^{3} + 876893 p^{3} T^{4} - 124 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 2457 T + 3055651 T^{2} + 2373006814 T^{3} + 3055651 p^{3} T^{4} + 2457 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 98 T + 312429 T^{2} + 544590556 T^{3} + 312429 p^{3} T^{4} + 98 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 454 T + 1115933 T^{2} - 211947628 T^{3} + 1115933 p^{3} T^{4} - 454 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 62 T + 1916783 T^{2} - 49066580 T^{3} + 1916783 p^{3} T^{4} - 62 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1730 T + 3077453 T^{2} + 3163337048 T^{3} + 3077453 p^{3} T^{4} + 1730 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−9.233931397370780496000734690379, −8.719243026852883946182323501749, −8.515219338558229886089153915703, −8.246294396157994482642149500102, −7.88201331211622300606667572475, −7.62705720594276076395134368002, −7.44944699937327579437091080795, −6.97367610852817562645295501066, −6.87926417767343449176060307862, −6.63875506682036162013318063733, −5.98830003359483019052754905898, −5.64095870881801335527927297915, −5.27264635589802897965813174786, −5.13884641291059655287867910592, −4.84067712662329460163497504988, −4.65358109970002137518156518792, −4.17992465916430655200616994267, −3.94572089865602352907953698463, −3.79111538006333546137614783122, −3.06329705782176075160176460293, −2.95906783236762031250215472187, −2.79678622655916191409153751102, −2.03187555941550636691050523068, −1.37552124714894316295151498380, −1.03748349726309080529707263037, 0, 0, 0,
1.03748349726309080529707263037, 1.37552124714894316295151498380, 2.03187555941550636691050523068, 2.79678622655916191409153751102, 2.95906783236762031250215472187, 3.06329705782176075160176460293, 3.79111538006333546137614783122, 3.94572089865602352907953698463, 4.17992465916430655200616994267, 4.65358109970002137518156518792, 4.84067712662329460163497504988, 5.13884641291059655287867910592, 5.27264635589802897965813174786, 5.64095870881801335527927297915, 5.98830003359483019052754905898, 6.63875506682036162013318063733, 6.87926417767343449176060307862, 6.97367610852817562645295501066, 7.44944699937327579437091080795, 7.62705720594276076395134368002, 7.88201331211622300606667572475, 8.246294396157994482642149500102, 8.515219338558229886089153915703, 8.719243026852883946182323501749, 9.233931397370780496000734690379
