L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s + 9-s − 4·13-s + 5·16-s + 17-s + 2·18-s + 8·19-s − 6·25-s − 8·26-s + 6·32-s + 2·34-s + 3·36-s + 16·38-s + 24·43-s − 14·49-s − 12·50-s − 12·52-s + 12·53-s + 24·59-s + 7·64-s − 24·67-s + 3·68-s + 4·72-s + 24·76-s + 81-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 1/3·9-s − 1.10·13-s + 5/4·16-s + 0.242·17-s + 0.471·18-s + 1.83·19-s − 6/5·25-s − 1.56·26-s + 1.06·32-s + 0.342·34-s + 1/2·36-s + 2.59·38-s + 3.65·43-s − 2·49-s − 1.69·50-s − 1.66·52-s + 1.64·53-s + 3.12·59-s + 7/8·64-s − 2.93·67-s + 0.363·68-s + 0.471·72-s + 2.75·76-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176868 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176868 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.248150726\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.248150726\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 17 | $C_1$ | \( 1 - T \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−9.547040289033150690294822441027, −8.613628586363707276957269542969, −7.958882066985845549410362974675, −7.37401646943555492580120617253, −7.36440720073199929079936076774, −6.73915837939670470281220727226, −5.97447010434826597680072551041, −5.51457882926074302977186886250, −5.32734843201609780311424178486, −4.47367404653766349933628761316, −4.13875090005906653574798019221, −3.49143456848613430686623125279, −2.76309813409348720224135008444, −2.28380910146500603557363711542, −1.19155942635042385588089355891,
1.19155942635042385588089355891, 2.28380910146500603557363711542, 2.76309813409348720224135008444, 3.49143456848613430686623125279, 4.13875090005906653574798019221, 4.47367404653766349933628761316, 5.32734843201609780311424178486, 5.51457882926074302977186886250, 5.97447010434826597680072551041, 6.73915837939670470281220727226, 7.36440720073199929079936076774, 7.37401646943555492580120617253, 7.958882066985845549410362974675, 8.613628586363707276957269542969, 9.547040289033150690294822441027