| L(s) = 1 | + 2·3-s + 3·9-s + 2·17-s − 8·23-s + 25-s + 4·27-s + 6·29-s − 8·43-s + 14·49-s + 4·51-s − 26·53-s + 30·61-s − 16·69-s + 2·75-s − 8·79-s + 5·81-s + 12·87-s − 6·101-s − 32·103-s − 40·107-s − 26·113-s + 22·121-s + 127-s − 16·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 0.485·17-s − 1.66·23-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 1.21·43-s + 2·49-s + 0.560·51-s − 3.57·53-s + 3.84·61-s − 1.92·69-s + 0.230·75-s − 0.900·79-s + 5/9·81-s + 1.28·87-s − 0.597·101-s − 3.15·103-s − 3.86·107-s − 2.44·113-s + 2·121-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16451136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.534162969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.534162969\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 137 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 190 T^{2} + p^{2} T^{4} \) |
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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
−8.425129584193236544597750577012, −8.189184634703033602973399759141, −8.093429542561579956402647341956, −7.75162725156854576490049156413, −7.05666276638728578100557202365, −6.78290486579060523317340953452, −6.73872052276482494096933791082, −5.88622136568464825008302966386, −5.82707782060081975069638774171, −5.16105916834161185473654892066, −4.87866693708687105734821872127, −4.30795016481760430238987107373, −3.91059215706329356940367123656, −3.72239281218278114928859809148, −3.12603792166247840551936944068, −2.56538218718704606359029221767, −2.50964986626511634023770713047, −1.61168947646378694768543394246, −1.43426421156518844878516825165, −0.48705822446828856356673527654,
0.48705822446828856356673527654, 1.43426421156518844878516825165, 1.61168947646378694768543394246, 2.50964986626511634023770713047, 2.56538218718704606359029221767, 3.12603792166247840551936944068, 3.72239281218278114928859809148, 3.91059215706329356940367123656, 4.30795016481760430238987107373, 4.87866693708687105734821872127, 5.16105916834161185473654892066, 5.82707782060081975069638774171, 5.88622136568464825008302966386, 6.73872052276482494096933791082, 6.78290486579060523317340953452, 7.05666276638728578100557202365, 7.75162725156854576490049156413, 8.093429542561579956402647341956, 8.189184634703033602973399759141, 8.425129584193236544597750577012
