L(s) = 1 | + 2·2-s + 2·4-s + 4·7-s + 16·11-s + 6·13-s + 8·14-s − 4·16-s − 14·17-s + 32·22-s + 4·23-s − 25·25-s + 12·26-s + 8·28-s + 104·31-s − 8·32-s − 28·34-s − 6·37-s + 16·41-s − 84·43-s + 32·44-s + 8·46-s + 36·47-s + 8·49-s − 50·50-s + 12·52-s − 106·53-s − 96·61-s + ⋯ |
L(s) = 1 | + 2-s + 1/2·4-s + 4/7·7-s + 1.45·11-s + 6/13·13-s + 4/7·14-s − 1/4·16-s − 0.823·17-s + 1.45·22-s + 4/23·23-s − 25-s + 6/13·26-s + 2/7·28-s + 3.35·31-s − 1/4·32-s − 0.823·34-s − 0.162·37-s + 0.390·41-s − 1.95·43-s + 8/11·44-s + 4/23·46-s + 0.765·47-s + 8/49·49-s − 50-s + 3/13·52-s − 2·53-s − 1.57·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.830195560\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.830195560\) |
\(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 18 T^{2} - 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )( 1 + 30 T + p^{2} T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 322 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 42 T + p^{2} T^{2} )( 1 + 42 T + p^{2} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 52 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 84 T + 3528 T^{2} + 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 36 T + 648 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 6562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 124 T + 7688 T^{2} - 124 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 28 T + p^{2} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 94 T + 4418 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 36 T + 648 T^{2} + 36 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 9442 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 126 T + 7938 T^{2} + 126 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−14.13431962902395914366131936347, −13.57316499754929552636352085392, −13.35405931161454968575998111711, −12.50323212053148142590062758371, −11.87787309647466466892041219742, −11.74529163801427426859045693301, −11.13697289022499561290794207190, −10.51396192154898461426291912661, −9.683088508157704384785666121113, −9.286388328972080110546009079123, −8.299684347999981198896119438282, −8.192394694420734508389404915960, −7.00971697305070295836234339632, −6.43122587510357414693953954050, −6.07898401272303230111741068109, −4.99702136963147333187798867256, −4.43971308324919386185934448900, −3.82730273436598180181313406219, −2.79394477055932692532199011707, −1.48102558081890001302219738510,
1.48102558081890001302219738510, 2.79394477055932692532199011707, 3.82730273436598180181313406219, 4.43971308324919386185934448900, 4.99702136963147333187798867256, 6.07898401272303230111741068109, 6.43122587510357414693953954050, 7.00971697305070295836234339632, 8.192394694420734508389404915960, 8.299684347999981198896119438282, 9.286388328972080110546009079123, 9.683088508157704384785666121113, 10.51396192154898461426291912661, 11.13697289022499561290794207190, 11.74529163801427426859045693301, 11.87787309647466466892041219742, 12.50323212053148142590062758371, 13.35405931161454968575998111711, 13.57316499754929552636352085392, 14.13431962902395914366131936347