| L(s) = 1 | − 3·2-s + 3-s + 6·4-s − 5-s − 3·6-s + 7-s − 10·8-s + 3·10-s + 11-s + 6·12-s − 13-s − 3·14-s − 15-s + 15·16-s − 17-s − 6·20-s + 21-s − 3·22-s − 10·24-s + 3·26-s + 6·28-s + 3·30-s − 21·32-s + 33-s + 3·34-s − 35-s − 37-s + ⋯ |
| L(s) = 1 | − 3·2-s + 3-s + 6·4-s − 5-s − 3·6-s + 7-s − 10·8-s + 3·10-s + 11-s + 6·12-s − 13-s − 3·14-s − 15-s + 15·16-s − 17-s − 6·20-s + 21-s − 3·22-s − 10·24-s + 3·26-s + 6·28-s + 3·30-s − 21·32-s + 33-s + 3·34-s − 35-s − 37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65939264 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65939264 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1924714678\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1924714678\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{3} \) |
| 101 | $C_1$ | \( ( 1 - T )^{3} \) |
| good | 3 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 5 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 7 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 11 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 13 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 17 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 59 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 67 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 83 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 97 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
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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
−10.22482014591979946604335099484, −9.726795063034602584647218766450, −9.661074496296445972701218916501, −9.296972861665135034752464824374, −8.845930916644655575057531817722, −8.673490439319629927427318148148, −8.628596496927416439142356134592, −8.124509847781570312948341235376, −7.989891165068875572869359213590, −7.74390877174625257861078268030, −7.18457248385764020162496399326, −7.03659879814863223377359885581, −6.98183756255354837071367620860, −6.38281776535415803391374192464, −6.02704123023410084160807175470, −5.65161334981991669304165218273, −5.00937812366992867608066973067, −4.63210002301672348639606858084, −3.92978293666613301173505521969, −3.45339367802752526624093070125, −3.21438967166661274010256641039, −2.55420776607694329291774055831, −2.03811030715590218381504477610, −1.96320419743726474061746195010, −1.00722729328853863224629721059,
1.00722729328853863224629721059, 1.96320419743726474061746195010, 2.03811030715590218381504477610, 2.55420776607694329291774055831, 3.21438967166661274010256641039, 3.45339367802752526624093070125, 3.92978293666613301173505521969, 4.63210002301672348639606858084, 5.00937812366992867608066973067, 5.65161334981991669304165218273, 6.02704123023410084160807175470, 6.38281776535415803391374192464, 6.98183756255354837071367620860, 7.03659879814863223377359885581, 7.18457248385764020162496399326, 7.74390877174625257861078268030, 7.989891165068875572869359213590, 8.124509847781570312948341235376, 8.628596496927416439142356134592, 8.673490439319629927427318148148, 8.845930916644655575057531817722, 9.296972861665135034752464824374, 9.661074496296445972701218916501, 9.726795063034602584647218766450, 10.22482014591979946604335099484
