| L(s) = 1 | + 2·2-s + 2·4-s − 2·7-s − 2·11-s + 12·13-s − 4·14-s − 4·16-s + 34·17-s − 4·22-s − 50·23-s − 25·25-s + 24·26-s − 4·28-s − 34·31-s − 8·32-s + 68·34-s − 36·37-s − 86·41-s − 60·43-s − 4·44-s − 100·46-s − 24·47-s + 2·49-s − 50·50-s + 24·52-s + 98·53-s − 120·61-s + ⋯ |
| L(s) = 1 | + 2-s + 1/2·4-s − 2/7·7-s − 0.181·11-s + 0.923·13-s − 2/7·14-s − 1/4·16-s + 2·17-s − 0.181·22-s − 2.17·23-s − 25-s + 0.923·26-s − 1/7·28-s − 1.09·31-s − 1/4·32-s + 2·34-s − 0.972·37-s − 2.09·41-s − 1.39·43-s − 0.0909·44-s − 2.17·46-s − 0.510·47-s + 2/49·49-s − 50-s + 6/13·52-s + 1.84·53-s − 1.96·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.983367289\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.983367289\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 50 T + 1250 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1657 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 17 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 43 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 60 T + 1800 T^{2} + 60 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 98 T + 4802 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6433 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 60 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 22 T + 242 T^{2} - 22 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 133 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 58 T + 1682 T^{2} + 58 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 11906 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 168 T + 14112 T^{2} + 168 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15793 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 216 T + 23328 T^{2} + 216 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−10.31286332128252370733620624606, −9.868586337381674866493586136947, −9.659817383487829228469807167286, −8.962888284368928598855520202544, −8.391943960915868506952733247852, −8.112130505861969170685556625039, −7.80456901990229461688007127132, −6.94377670737231489421503326078, −6.92242254240119981523356035750, −6.00296021380477886825535588145, −5.88967647929635056278981291938, −5.42495862893767903987358342022, −5.03147831135004752490207112361, −4.25642919433004512373981629020, −3.75648045513992328768536203312, −3.43033626718345635446302121712, −3.09008648414836670765458307857, −1.84222498900287373909741836402, −1.80277634430581520046007487771, −0.47628053682952386853819826091,
0.47628053682952386853819826091, 1.80277634430581520046007487771, 1.84222498900287373909741836402, 3.09008648414836670765458307857, 3.43033626718345635446302121712, 3.75648045513992328768536203312, 4.25642919433004512373981629020, 5.03147831135004752490207112361, 5.42495862893767903987358342022, 5.88967647929635056278981291938, 6.00296021380477886825535588145, 6.92242254240119981523356035750, 6.94377670737231489421503326078, 7.80456901990229461688007127132, 8.112130505861969170685556625039, 8.391943960915868506952733247852, 8.962888284368928598855520202544, 9.659817383487829228469807167286, 9.868586337381674866493586136947, 10.31286332128252370733620624606
