| L(s) = 1 | − 15·5-s + 4·7-s + 5·11-s + 7·13-s − 155·17-s + 50·19-s + 285·23-s + 150·25-s − 115·29-s + 115·31-s − 60·35-s − 384·37-s − 580·41-s + 797·43-s − 145·47-s − 218·49-s + 400·53-s − 75·55-s + 380·59-s − 152·61-s − 105·65-s − 2·67-s + 40·71-s − 980·73-s + 20·77-s − 1.01e3·79-s + 270·83-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.215·7-s + 0.137·11-s + 0.149·13-s − 2.21·17-s + 0.603·19-s + 2.58·23-s + 6/5·25-s − 0.736·29-s + 0.666·31-s − 0.289·35-s − 1.70·37-s − 2.20·41-s + 2.82·43-s − 0.450·47-s − 0.635·49-s + 1.03·53-s − 0.183·55-s + 0.838·59-s − 0.319·61-s − 0.200·65-s − 0.00364·67-s + 0.0668·71-s − 1.57·73-s + 0.0296·77-s − 1.44·79-s + 0.357·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{9} \cdot 5^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + p T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 - 4 T + 234 T^{2} + 5554 T^{3} + 234 p^{3} T^{4} - 4 p^{6} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 5 T + 1105 T^{2} + 17950 T^{3} + 1105 p^{3} T^{4} - 5 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 7 T + 5822 T^{2} - 37183 T^{3} + 5822 p^{3} T^{4} - 7 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 155 T + 20347 T^{2} + 1564790 T^{3} + 20347 p^{3} T^{4} + 155 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 - 50 T + 206 p T^{2} - 317888 T^{3} + 206 p^{4} T^{4} - 50 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 285 T + 52701 T^{2} - 6381690 T^{3} + 52701 p^{3} T^{4} - 285 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 115 T + 25759 T^{2} - 830870 T^{3} + 25759 p^{3} T^{4} + 115 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 - 115 T + 60141 T^{2} - 5913626 T^{3} + 60141 p^{3} T^{4} - 115 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 384 T + 84036 T^{2} + 16234306 T^{3} + 84036 p^{3} T^{4} + 384 p^{6} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 + 580 T + 296575 T^{2} + 83865640 T^{3} + 296575 p^{3} T^{4} + 580 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 797 T + 381041 T^{2} - 121376222 T^{3} + 381041 p^{3} T^{4} - 797 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 145 T + 62317 T^{2} + 44496910 T^{3} + 62317 p^{3} T^{4} + 145 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 400 T + 388459 T^{2} - 106443280 T^{3} + 388459 p^{3} T^{4} - 400 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 380 T + 588745 T^{2} - 150882920 T^{3} + 588745 p^{3} T^{4} - 380 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 152 T + 593468 T^{2} + 63933158 T^{3} + 593468 p^{3} T^{4} + 152 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 + 2 T + 658290 T^{2} - 19566248 T^{3} + 658290 p^{3} T^{4} + 2 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 - 40 T + 396361 T^{2} + 187438400 T^{3} + 396361 p^{3} T^{4} - 40 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 980 T + 1431584 T^{2} + 778921274 T^{3} + 1431584 p^{3} T^{4} + 980 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 1013 T + 1531332 T^{2} + 908300089 T^{3} + 1531332 p^{3} T^{4} + 1013 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 270 T + 1413933 T^{2} - 224225820 T^{3} + 1413933 p^{3} T^{4} - 270 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 1020 T + 2161707 T^{2} + 1313072760 T^{3} + 2161707 p^{3} T^{4} + 1020 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 - 720 T + 2703456 T^{2} - 1286818562 T^{3} + 2703456 p^{3} T^{4} - 720 p^{6} T^{5} + p^{9} T^{6} \) |
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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
−8.203911531670464759940125026350, −7.64380663976341507378394020775, −7.55946997231154401900227337073, −7.28835848012427033337765984248, −7.00534738883418576579330847413, −6.94225432547727477007322649185, −6.62698754221920298737987270576, −6.39218483045788375235531352334, −5.93351718786442928232447153798, −5.69961693841825594886236931657, −5.25709546793665623471513887872, −5.02344678502464563134252198876, −4.94092016338701379526725093915, −4.41227933334882806459087479621, −4.27655146000530921664273408744, −4.14922912369969494526895989867, −3.42064221209767509242966521354, −3.35531211753086213157463260978, −3.33407709022172054627372905159, −2.55573375418553993654401502698, −2.38833519190395985673769927297, −2.21854426906670479878243831330, −1.32022018524105768834340261031, −1.20391563311979249016700455810, −1.05290004974472659833071676206, 0, 0, 0,
1.05290004974472659833071676206, 1.20391563311979249016700455810, 1.32022018524105768834340261031, 2.21854426906670479878243831330, 2.38833519190395985673769927297, 2.55573375418553993654401502698, 3.33407709022172054627372905159, 3.35531211753086213157463260978, 3.42064221209767509242966521354, 4.14922912369969494526895989867, 4.27655146000530921664273408744, 4.41227933334882806459087479621, 4.94092016338701379526725093915, 5.02344678502464563134252198876, 5.25709546793665623471513887872, 5.69961693841825594886236931657, 5.93351718786442928232447153798, 6.39218483045788375235531352334, 6.62698754221920298737987270576, 6.94225432547727477007322649185, 7.00534738883418576579330847413, 7.28835848012427033337765984248, 7.55946997231154401900227337073, 7.64380663976341507378394020775, 8.203911531670464759940125026350
