| 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
| \(L(2)\) |
\(\approx\) |
\(9.747727675\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.747727675\) |
| \(L(\frac{5}{2})\) |
|
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
−7.80070970887412735482762972419, −7.34113304741699661952286298249, −7.26397444128624036873231404258, −7.14102993843558305942953036009, −6.34320096136471952322337072524, −6.18203592365138161002893201068, −6.17325933171763616436865095073, −5.74068739919378833234893708673, −5.65410904067746767851249594977, −5.59576874861547462220902075233, −4.84931665504316128374340518764, −4.73877211429968613434398865026, −4.59140837163798799139414204620, −4.08685069583425455750513953733, −3.78392473397953124102140925707, −3.49248635657990941030930276291, −3.00934274402454739849884505002, −2.92967343352485972025345683551, −2.53931372010116937766953503282, −1.95647970699696974840266089680, −1.93172305014137712761238472641, −1.50661475474859047616749923635, −1.04091342433469792579591128801, −0.74086958051538614133509596372, −0.42051496302280679173631000979,
0.42051496302280679173631000979, 0.74086958051538614133509596372, 1.04091342433469792579591128801, 1.50661475474859047616749923635, 1.93172305014137712761238472641, 1.95647970699696974840266089680, 2.53931372010116937766953503282, 2.92967343352485972025345683551, 3.00934274402454739849884505002, 3.49248635657990941030930276291, 3.78392473397953124102140925707, 4.08685069583425455750513953733, 4.59140837163798799139414204620, 4.73877211429968613434398865026, 4.84931665504316128374340518764, 5.59576874861547462220902075233, 5.65410904067746767851249594977, 5.74068739919378833234893708673, 6.17325933171763616436865095073, 6.18203592365138161002893201068, 6.34320096136471952322337072524, 7.14102993843558305942953036009, 7.26397444128624036873231404258, 7.34113304741699661952286298249, 7.80070970887412735482762972419
