L(s) = 1 | − 2·4-s − 2·7-s − 5·9-s + 4·16-s + 12·17-s + 12·23-s − 10·25-s + 4·28-s − 8·31-s + 10·36-s − 18·41-s + 6·47-s − 11·49-s + 10·63-s − 8·64-s − 24·68-s − 30·71-s + 22·73-s − 20·79-s + 16·81-s + 12·89-s − 24·92-s + 16·97-s + 20·100-s − 8·103-s − 8·112-s − 12·113-s + ⋯ |
L(s) = 1 | − 4-s − 0.755·7-s − 5/3·9-s + 16-s + 2.91·17-s + 2.50·23-s − 2·25-s + 0.755·28-s − 1.43·31-s + 5/3·36-s − 2.81·41-s + 0.875·47-s − 1.57·49-s + 1.25·63-s − 64-s − 2.91·68-s − 3.56·71-s + 2.57·73-s − 2.25·79-s + 16/9·81-s + 1.27·89-s − 2.50·92-s + 1.62·97-s + 2·100-s − 0.788·103-s − 0.755·112-s − 1.12·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−9.348654663657790611777830759538, −9.032345851694468357832029856913, −8.487594403486838813155108109695, −7.953080976467459568741615470163, −7.59911177067371416247669368567, −6.92702526110685059626904983928, −6.13026790010533892031415629049, −5.52397436821147602871559735263, −5.44973416215471530225317007593, −4.81124860316907390767416903100, −3.50910294340479626549928321873, −3.49656973901712201010716232454, −2.90414738025441870892963779928, −1.39370194281586028856128276808, 0,
1.39370194281586028856128276808, 2.90414738025441870892963779928, 3.49656973901712201010716232454, 3.50910294340479626549928321873, 4.81124860316907390767416903100, 5.44973416215471530225317007593, 5.52397436821147602871559735263, 6.13026790010533892031415629049, 6.92702526110685059626904983928, 7.59911177067371416247669368567, 7.953080976467459568741615470163, 8.487594403486838813155108109695, 9.032345851694468357832029856913, 9.348654663657790611777830759538