| L(s) = 1 | + 4-s + 4·7-s − 6·9-s − 3·16-s + 4·17-s − 12·23-s + 4·28-s − 6·36-s + 12·43-s + 4·47-s − 2·49-s − 20·61-s − 24·63-s − 7·64-s + 4·68-s − 28·73-s + 27·81-s − 28·83-s − 12·92-s − 20·101-s − 12·112-s + 16·119-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.51·7-s − 2·9-s − 3/4·16-s + 0.970·17-s − 2.50·23-s + 0.755·28-s − 36-s + 1.82·43-s + 0.583·47-s − 2/7·49-s − 2.56·61-s − 3.02·63-s − 7/8·64-s + 0.485·68-s − 3.27·73-s + 3·81-s − 3.07·83-s − 1.25·92-s − 1.99·101-s − 1.13·112-s + 1.46·119-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81450625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 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
−7.81208363530236726404997665037, −7.38116270924417067237467240847, −6.93025724488558533499758945176, −6.39921927696296215619010693838, −6.09651207259395221177883347994, −5.75147952375285553469512159742, −5.65872611871033374739293143102, −5.33126707827038524156288453971, −4.59405259041441511593151909866, −4.57098776577191378324447689144, −4.08284126326574961570411921192, −3.71180799570969580182241767788, −3.03820340865098716968356653887, −2.73742954127320276525294729056, −2.58196722155030030635340973388, −1.84623628770838907716813073375, −1.66595214228068149650120013126, −1.12436510392985989269873786035, 0, 0,
1.12436510392985989269873786035, 1.66595214228068149650120013126, 1.84623628770838907716813073375, 2.58196722155030030635340973388, 2.73742954127320276525294729056, 3.03820340865098716968356653887, 3.71180799570969580182241767788, 4.08284126326574961570411921192, 4.57098776577191378324447689144, 4.59405259041441511593151909866, 5.33126707827038524156288453971, 5.65872611871033374739293143102, 5.75147952375285553469512159742, 6.09651207259395221177883347994, 6.39921927696296215619010693838, 6.93025724488558533499758945176, 7.38116270924417067237467240847, 7.81208363530236726404997665037
