L(s) = 1 | + 3-s + 8·7-s + 3·9-s − 6·11-s − 2·13-s + 6·17-s + 7·19-s + 8·21-s − 6·23-s + 5·25-s + 8·27-s − 4·31-s − 6·33-s − 20·37-s − 2·39-s − 9·41-s − 4·43-s + 34·49-s + 6·51-s − 6·53-s + 7·57-s − 9·59-s + 4·61-s + 24·63-s − 7·67-s − 6·69-s − 6·71-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 3.02·7-s + 9-s − 1.80·11-s − 0.554·13-s + 1.45·17-s + 1.60·19-s + 1.74·21-s − 1.25·23-s + 25-s + 1.53·27-s − 0.718·31-s − 1.04·33-s − 3.28·37-s − 0.320·39-s − 1.40·41-s − 0.609·43-s + 34/7·49-s + 0.840·51-s − 0.824·53-s + 0.927·57-s − 1.17·59-s + 0.512·61-s + 3.02·63-s − 0.855·67-s − 0.722·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.562255824\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.562255824\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 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 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−11.79046945920849194084307987192, −11.70267975460397978759753019787, −10.73359721735879179389445418935, −10.71847297110403078461802098912, −10.03159301912368317158478020777, −9.932640092240573728227139780063, −8.879975529068642090755983407217, −8.464498472816492501032971920398, −8.064793730236352487603491001056, −7.81492517044600848278144870102, −7.18390465856659408311806978343, −7.12523186046341795950573104520, −5.56460508320998101200922413237, −5.37100781141545302116005837049, −4.80639296637173689723380906051, −4.66394074691694157525008378116, −3.49310160595025062941920708005, −2.90543307958866180738477720057, −1.74890550306580313603717464941, −1.55470481856806376749536800444,
1.55470481856806376749536800444, 1.74890550306580313603717464941, 2.90543307958866180738477720057, 3.49310160595025062941920708005, 4.66394074691694157525008378116, 4.80639296637173689723380906051, 5.37100781141545302116005837049, 5.56460508320998101200922413237, 7.12523186046341795950573104520, 7.18390465856659408311806978343, 7.81492517044600848278144870102, 8.064793730236352487603491001056, 8.464498472816492501032971920398, 8.879975529068642090755983407217, 9.932640092240573728227139780063, 10.03159301912368317158478020777, 10.71847297110403078461802098912, 10.73359721735879179389445418935, 11.70267975460397978759753019787, 11.79046945920849194084307987192