L(s) = 1 | + 2-s − 3·5-s + 7-s + 3·9-s − 3·10-s − 11-s + 14-s + 17-s + 3·18-s − 19-s − 22-s + 6·25-s − 29-s + 34-s − 3·35-s − 38-s − 41-s + 43-s − 9·45-s + 6·50-s + 3·55-s − 58-s − 61-s + 3·63-s + 67-s − 3·70-s + 73-s + ⋯ |
L(s) = 1 | + 2-s − 3·5-s + 7-s + 3·9-s − 3·10-s − 11-s + 14-s + 17-s + 3·18-s − 19-s − 22-s + 6·25-s − 29-s + 34-s − 3·35-s − 38-s − 41-s + 43-s − 9·45-s + 6·50-s + 3·55-s − 58-s − 61-s + 3·63-s + 67-s − 3·70-s + 73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 107^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{3} \cdot 107^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8138596026\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8138596026\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | $C_1$ | \( ( 1 + T )^{3} \) |
| 107 | $C_1$ | \( ( 1 + T )^{3} \) |
good | 2 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 17 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 19 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 29 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 41 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 43 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 61 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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 | $C_6$ | \( 1 - T + T^{2} - T^{3} + T^{4} - T^{5} + T^{6} \) |
| 79 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{3}( 1 + T )^{3} \) |
| 89 | $C_6$ | \( 1 + T + T^{2} + T^{3} + T^{4} + T^{5} + T^{6} \) |
| 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
−10.26982506078357238541825753478, −9.467994169589191855473599328590, −9.324141254380461169760587001755, −8.972750987382410267705541612372, −8.490467274549382096495902889743, −8.047008761313330853147939240620, −7.956946056206819531068158383000, −7.81940721043480553119510044341, −7.53313432835816202376978315834, −7.15506779281197965351716475533, −7.03494875052797441026295742556, −6.53528954375084791442464160975, −6.40850087658375716092715946670, −5.34677270563315951793799564245, −5.29409387424004825957703162325, −4.94179386008269118469647978069, −4.46032078340841994428082786354, −4.45926733662171269660191710541, −4.09954223669269929260750447000, −3.72280023376165040809829371667, −3.70304233461520988126906660880, −2.99693726393766606120851860447, −2.42326741041247151086863629066, −1.59394947702005839200279605791, −1.15100878527600699774350136015,
1.15100878527600699774350136015, 1.59394947702005839200279605791, 2.42326741041247151086863629066, 2.99693726393766606120851860447, 3.70304233461520988126906660880, 3.72280023376165040809829371667, 4.09954223669269929260750447000, 4.45926733662171269660191710541, 4.46032078340841994428082786354, 4.94179386008269118469647978069, 5.29409387424004825957703162325, 5.34677270563315951793799564245, 6.40850087658375716092715946670, 6.53528954375084791442464160975, 7.03494875052797441026295742556, 7.15506779281197965351716475533, 7.53313432835816202376978315834, 7.81940721043480553119510044341, 7.956946056206819531068158383000, 8.047008761313330853147939240620, 8.490467274549382096495902889743, 8.972750987382410267705541612372, 9.324141254380461169760587001755, 9.467994169589191855473599328590, 10.26982506078357238541825753478