| L(s) = 1 | + 2·2-s + 4-s + 2·7-s + 2·11-s + 4·14-s + 16-s + 2·17-s + 10·19-s + 4·22-s + 2·23-s + 2·28-s − 8·29-s − 2·32-s + 4·34-s + 6·37-s + 20·38-s + 2·41-s + 12·43-s + 2·44-s + 4·46-s + 2·47-s − 9·49-s + 4·53-s − 16·58-s − 22·59-s + 12·61-s − 11·64-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 0.755·7-s + 0.603·11-s + 1.06·14-s + 1/4·16-s + 0.485·17-s + 2.29·19-s + 0.852·22-s + 0.417·23-s + 0.377·28-s − 1.48·29-s − 0.353·32-s + 0.685·34-s + 0.986·37-s + 3.24·38-s + 0.312·41-s + 1.82·43-s + 0.301·44-s + 0.589·46-s + 0.291·47-s − 9/7·49-s + 0.549·53-s − 2.10·58-s − 2.86·59-s + 1.53·61-s − 1.37·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.197432990\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.197432990\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2 T + 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 10 T + 61 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 8 T + 66 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 75 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 90 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 23 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 138 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 10 T + 159 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 18 T + 221 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T + 150 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 195 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−9.042108873220691970721145327688, −9.038303246615674129741912420181, −8.047789792445424609641471308052, −7.78903571165956974370921895178, −7.74193408000098379774415786049, −7.21133434994057695213254037416, −6.74719232777622347283862389862, −6.28788355886798183937497448502, −5.67279215383912692861881256720, −5.60139774587708668935289470370, −5.05311411854726103041484059790, −4.91449834580153901885306396802, −4.30585944351200599425917937624, −3.98889211675823804481276091346, −3.47813171042685847801851810315, −3.25409295182062201412747301077, −2.58177776637967524473544864204, −1.94093501066343588614189941391, −1.29914144667592596561329551750, −0.798073002033207128150251416184,
0.798073002033207128150251416184, 1.29914144667592596561329551750, 1.94093501066343588614189941391, 2.58177776637967524473544864204, 3.25409295182062201412747301077, 3.47813171042685847801851810315, 3.98889211675823804481276091346, 4.30585944351200599425917937624, 4.91449834580153901885306396802, 5.05311411854726103041484059790, 5.60139774587708668935289470370, 5.67279215383912692861881256720, 6.28788355886798183937497448502, 6.74719232777622347283862389862, 7.21133434994057695213254037416, 7.74193408000098379774415786049, 7.78903571165956974370921895178, 8.047789792445424609641471308052, 9.038303246615674129741912420181, 9.042108873220691970721145327688
