| L(s) = 1 | − 4·2-s + 8·4-s + 2·5-s − 8·8-s − 5·9-s − 8·10-s + 2·11-s − 4·16-s − 4·17-s + 20·18-s + 16·20-s − 8·22-s − 7·25-s + 32·32-s + 16·34-s − 40·36-s − 16·40-s + 16·44-s − 10·45-s + 16·47-s − 10·49-s + 28·50-s − 12·53-s + 4·55-s − 64·64-s − 14·67-s − 32·68-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 4·4-s + 0.894·5-s − 2.82·8-s − 5/3·9-s − 2.52·10-s + 0.603·11-s − 16-s − 0.970·17-s + 4.71·18-s + 3.57·20-s − 1.70·22-s − 7/5·25-s + 5.65·32-s + 2.74·34-s − 6.66·36-s − 2.52·40-s + 2.41·44-s − 1.49·45-s + 2.33·47-s − 1.42·49-s + 3.95·50-s − 1.64·53-s + 0.539·55-s − 8·64-s − 1.71·67-s − 3.88·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 958441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 958441 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1707542378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1707542378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 89 | $C_2$ | \( 1 - 15 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.558580597072866362021424590520, −7.80306712748985717398688888740, −7.52305447162640040957017397420, −7.20685073138161009710557405792, −6.36261389471308870138602900888, −6.22751027744569053248642726916, −5.85551354442718043458262245689, −5.05232126574041880500064099657, −4.54859976719023207502696657866, −3.88151059584158277500212230098, −2.98547319486052976896224337133, −2.35691778866774912809470218958, −1.93214884663950861864873375229, −1.33739722426120305081140732346, −0.31591320242433503945261590043,
0.31591320242433503945261590043, 1.33739722426120305081140732346, 1.93214884663950861864873375229, 2.35691778866774912809470218958, 2.98547319486052976896224337133, 3.88151059584158277500212230098, 4.54859976719023207502696657866, 5.05232126574041880500064099657, 5.85551354442718043458262245689, 6.22751027744569053248642726916, 6.36261389471308870138602900888, 7.20685073138161009710557405792, 7.52305447162640040957017397420, 7.80306712748985717398688888740, 8.558580597072866362021424590520
