| L(s) = 1 | + 2·5-s − 8·13-s − 4·17-s + 3·25-s − 12·29-s − 16·37-s − 16·41-s + 6·49-s − 12·53-s − 20·61-s − 16·65-s + 12·73-s − 8·85-s − 8·89-s + 4·97-s + 20·101-s + 12·109-s − 28·113-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s − 24·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 2.21·13-s − 0.970·17-s + 3/5·25-s − 2.22·29-s − 2.63·37-s − 2.49·41-s + 6/7·49-s − 1.64·53-s − 2.56·61-s − 1.98·65-s + 1.40·73-s − 0.867·85-s − 0.847·89-s + 0.406·97-s + 1.99·101-s + 1.14·109-s − 2.63·113-s − 0.181·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.99·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.736672834619132221134147527274, −8.225638028697688497278841291987, −7.72822995983623661564723283393, −7.37118007557131887298768364978, −7.12618253426463859043892115223, −6.69287551516549112391053146887, −6.41133730696608170877034931075, −5.86206138543870870604177612763, −5.33033869973738081391347296256, −5.19492125744564010499942754511, −4.74510164876638534729548892448, −4.46560681954322177437005313439, −3.62866238101383552945045052083, −3.38371142387337403879635197901, −2.80789565732270295599809557896, −2.14218903137690434371926252108, −1.94744004862543050759831160183, −1.49670449859451459272590980544, 0, 0,
1.49670449859451459272590980544, 1.94744004862543050759831160183, 2.14218903137690434371926252108, 2.80789565732270295599809557896, 3.38371142387337403879635197901, 3.62866238101383552945045052083, 4.46560681954322177437005313439, 4.74510164876638534729548892448, 5.19492125744564010499942754511, 5.33033869973738081391347296256, 5.86206138543870870604177612763, 6.41133730696608170877034931075, 6.69287551516549112391053146887, 7.12618253426463859043892115223, 7.37118007557131887298768364978, 7.72822995983623661564723283393, 8.225638028697688497278841291987, 8.736672834619132221134147527274
