| L(s) = 1 | − 3·3-s + 3·5-s − 2·7-s + 6·9-s − 6·11-s − 2·13-s − 9·15-s + 6·21-s − 2·23-s + 6·25-s − 10·27-s + 8·29-s − 3·31-s + 18·33-s − 6·35-s + 6·39-s + 8·41-s + 10·43-s + 18·45-s + 6·47-s − 15·49-s − 18·55-s − 14·59-s − 2·61-s − 12·63-s − 6·65-s + 16·67-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s − 0.755·7-s + 2·9-s − 1.80·11-s − 0.554·13-s − 2.32·15-s + 1.30·21-s − 0.417·23-s + 6/5·25-s − 1.92·27-s + 1.48·29-s − 0.538·31-s + 3.13·33-s − 1.01·35-s + 0.960·39-s + 1.24·41-s + 1.52·43-s + 2.68·45-s + 0.875·47-s − 2.14·49-s − 2.42·55-s − 1.82·59-s − 0.256·61-s − 1.51·63-s − 0.744·65-s + 1.95·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{3} \cdot 31^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 5 | $C_1$ | \( ( 1 - T )^{3} \) |
| 31 | $C_1$ | \( ( 1 + T )^{3} \) |
| good | 7 | $S_4\times C_2$ | \( 1 + 2 T + 19 T^{2} + 26 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{3} \) |
| 13 | $S_4\times C_2$ | \( 1 + 2 T + 17 T^{2} + 54 T^{3} + 17 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 23 T^{2} + 52 T^{3} + 23 p T^{4} + p^{3} T^{6} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{3} \) |
| 23 | $S_4\times C_2$ | \( 1 + 2 T + 65 T^{2} + 88 T^{3} + 65 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 8 T + 71 T^{2} - 334 T^{3} + 71 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 7 T^{2} - 358 T^{3} - 7 p T^{4} + p^{3} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 8 T + 91 T^{2} - 528 T^{3} + 91 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 10 T + 125 T^{2} - 852 T^{3} + 125 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 6 T + 105 T^{2} - 496 T^{3} + 105 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 119 T^{2} + 76 T^{3} + 119 p T^{4} + p^{3} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 14 T + 149 T^{2} + 982 T^{3} + 149 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 2 T + 147 T^{2} + 140 T^{3} + 147 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 16 T + 203 T^{2} - 1842 T^{3} + 203 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 20 T + 297 T^{2} + 2706 T^{3} + 297 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 149 T^{2} + 86 T^{3} + 149 p T^{4} + p^{3} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 8 T + 145 T^{2} + 884 T^{3} + 145 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 2 T + 117 T^{2} + 336 T^{3} + 117 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 6 T + 83 T^{2} - 2 T^{3} + 83 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \) |
| 97 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{3} \) |
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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
−7.41963951397145621851859305053, −6.81332001554274508882346098847, −6.78060700561801050739994293033, −6.70406668294656445826128028787, −6.18429280038503082016875287505, −6.10688325980426750387041046262, −6.02638042413564826646118332979, −5.59842225371445113930707612820, −5.51118090761272966970292732896, −5.38661539576505188699813364066, −4.90932484517098327126495648645, −4.78954391020416043053887302641, −4.77798237426746562115645824936, −4.19216974095262438641336417606, −4.09347046177356715677228112139, −3.87108106333325805092640681357, −3.24780189096509911922560865760, −2.99482942769004195950517569942, −2.84299906812264170911575751143, −2.51663649653675136050709525526, −2.20896888689304046289931564222, −2.12464417045084418329065396411, −1.26933595284785679364579699419, −1.26469010717315475648564022406, −1.08991248389240607932263157560, 0, 0, 0,
1.08991248389240607932263157560, 1.26469010717315475648564022406, 1.26933595284785679364579699419, 2.12464417045084418329065396411, 2.20896888689304046289931564222, 2.51663649653675136050709525526, 2.84299906812264170911575751143, 2.99482942769004195950517569942, 3.24780189096509911922560865760, 3.87108106333325805092640681357, 4.09347046177356715677228112139, 4.19216974095262438641336417606, 4.77798237426746562115645824936, 4.78954391020416043053887302641, 4.90932484517098327126495648645, 5.38661539576505188699813364066, 5.51118090761272966970292732896, 5.59842225371445113930707612820, 6.02638042413564826646118332979, 6.10688325980426750387041046262, 6.18429280038503082016875287505, 6.70406668294656445826128028787, 6.78060700561801050739994293033, 6.81332001554274508882346098847, 7.41963951397145621851859305053
