| L(s) = 1 | − 3·2-s + 4·4-s − 3·5-s − 3·8-s + 9·10-s + 3·11-s − 3·13-s + 3·16-s + 6·17-s − 12·20-s − 9·22-s + 9·23-s + 5·25-s + 9·26-s + 9·29-s + 6·31-s − 6·32-s − 18·34-s + 14·37-s + 9·40-s − 3·41-s − 43-s + 12·44-s − 27·46-s − 15·50-s − 12·52-s − 9·55-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 2·4-s − 1.34·5-s − 1.06·8-s + 2.84·10-s + 0.904·11-s − 0.832·13-s + 3/4·16-s + 1.45·17-s − 2.68·20-s − 1.91·22-s + 1.87·23-s + 25-s + 1.76·26-s + 1.67·29-s + 1.07·31-s − 1.06·32-s − 3.08·34-s + 2.30·37-s + 1.42·40-s − 0.468·41-s − 0.152·43-s + 1.80·44-s − 3.98·46-s − 2.12·50-s − 1.66·52-s − 1.21·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5945267191\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5945267191\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + 5 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 3 T + 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 9 T + 50 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 56 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 43 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 24 T + 253 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 15 T + 142 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 3 T + 100 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
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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
−9.727759499055594692073227863741, −9.379483821677336249419715457555, −8.947535624487188240815107163715, −8.749954109936846503630727204663, −8.092664724362537143593313351415, −8.077373642704422655210598532052, −7.43853907270080148475361540732, −7.42871854014678518750790384688, −6.85761923718958144191156678633, −6.14697644250159732574700058656, −6.11514686055047604524358271980, −5.01439142857461558459354154347, −4.76742643942536813712270095210, −4.33351894071080335503581867442, −3.57232428053032053845843337126, −3.02544826139410888308776915015, −2.79927329427824750891711867144, −1.62912832098938095890362177236, −0.854439487741414738994765056893, −0.71500336246337981501250206205,
0.71500336246337981501250206205, 0.854439487741414738994765056893, 1.62912832098938095890362177236, 2.79927329427824750891711867144, 3.02544826139410888308776915015, 3.57232428053032053845843337126, 4.33351894071080335503581867442, 4.76742643942536813712270095210, 5.01439142857461558459354154347, 6.11514686055047604524358271980, 6.14697644250159732574700058656, 6.85761923718958144191156678633, 7.42871854014678518750790384688, 7.43853907270080148475361540732, 8.077373642704422655210598532052, 8.092664724362537143593313351415, 8.749954109936846503630727204663, 8.947535624487188240815107163715, 9.379483821677336249419715457555, 9.727759499055594692073227863741
