| L(s) = 1 | + 4·2-s − 3-s + 8·4-s − 4·6-s + 6·7-s + 8·8-s + 9-s − 8·12-s + 24·14-s − 4·16-s + 4·18-s − 19-s − 6·21-s − 8·24-s − 9·25-s − 27-s + 48·28-s + 20·29-s − 32·32-s + 8·36-s − 4·38-s + 16·41-s − 24·42-s − 2·43-s + 4·48-s + 13·49-s − 36·50-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 0.577·3-s + 4·4-s − 1.63·6-s + 2.26·7-s + 2.82·8-s + 1/3·9-s − 2.30·12-s + 6.41·14-s − 16-s + 0.942·18-s − 0.229·19-s − 1.30·21-s − 1.63·24-s − 9/5·25-s − 0.192·27-s + 9.07·28-s + 3.71·29-s − 5.65·32-s + 4/3·36-s − 0.648·38-s + 2.49·41-s − 3.70·42-s − 0.304·43-s + 0.577·48-s + 13/7·49-s − 5.09·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185193 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.280178051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.280178051\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 + T \) |
| 19 | $C_1$ | \( 1 + T \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.939322066982885493110877308795, −8.513296718878561122779255009178, −8.150338074826865833544092224319, −7.33179869161333338136909211767, −7.06801910593054313669065331224, −6.12831923497636436478284774657, −6.02567308265222601671827257647, −5.52130085797761183807247002982, −4.80392893944635697455333037352, −4.75227634922127063518685270995, −4.20883425493377341015603476651, −3.90431057417865746907593992810, −2.76056898471702515117641585129, −2.44031011733442394942262196334, −1.37927155401612470737240964655,
1.37927155401612470737240964655, 2.44031011733442394942262196334, 2.76056898471702515117641585129, 3.90431057417865746907593992810, 4.20883425493377341015603476651, 4.75227634922127063518685270995, 4.80392893944635697455333037352, 5.52130085797761183807247002982, 6.02567308265222601671827257647, 6.12831923497636436478284774657, 7.06801910593054313669065331224, 7.33179869161333338136909211767, 8.150338074826865833544092224319, 8.513296718878561122779255009178, 8.939322066982885493110877308795
