| L(s) = 1 | − 2-s − 3-s + 5-s + 6-s − 2·7-s + 8-s − 10-s − 6·11-s − 2·13-s + 2·14-s − 15-s − 16-s + 6·17-s − 7·19-s + 2·21-s + 6·22-s − 9·23-s − 24-s + 2·26-s + 27-s + 30-s − 8·31-s + 6·33-s − 6·34-s − 2·35-s + 10·37-s + 7·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.755·7-s + 0.353·8-s − 0.316·10-s − 1.80·11-s − 0.554·13-s + 0.534·14-s − 0.258·15-s − 1/4·16-s + 1.45·17-s − 1.60·19-s + 0.436·21-s + 1.27·22-s − 1.87·23-s − 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.182·30-s − 1.43·31-s + 1.04·33-s − 1.02·34-s − 0.338·35-s + 1.64·37-s + 1.13·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2868396421\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2868396421\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 58 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 4 T - 57 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 15 T + 136 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.94873816678968340129075164655, −10.16638017014507483516433699708, −10.09003637158575800221049725533, −9.889088692994453425513456718583, −9.402657941259542503965876920269, −8.750039238270802997566335987055, −8.160784941207707021441747817290, −7.993120986924548534762000072924, −7.51857904000476600701679172902, −6.96557143561581609103695248426, −6.38873088312627976819022714305, −5.84590951473238843458065623410, −5.50777839834613963862777190539, −5.21298337763800090190969209635, −4.11896752997089085795338377348, −4.07941694382071114517495335772, −2.83036909704991749660056451745, −2.54302185568816031686730999018, −1.68812285603811603991036572624, −0.34306327853343236205424443068,
0.34306327853343236205424443068, 1.68812285603811603991036572624, 2.54302185568816031686730999018, 2.83036909704991749660056451745, 4.07941694382071114517495335772, 4.11896752997089085795338377348, 5.21298337763800090190969209635, 5.50777839834613963862777190539, 5.84590951473238843458065623410, 6.38873088312627976819022714305, 6.96557143561581609103695248426, 7.51857904000476600701679172902, 7.993120986924548534762000072924, 8.160784941207707021441747817290, 8.750039238270802997566335987055, 9.402657941259542503965876920269, 9.889088692994453425513456718583, 10.09003637158575800221049725533, 10.16638017014507483516433699708, 10.94873816678968340129075164655
