| L(s) = 1 | + 3-s + 4-s + 2·7-s + 9-s + 12-s − 2·13-s + 16-s − 8·19-s + 2·21-s − 6·25-s + 27-s + 2·28-s − 16·31-s + 36-s + 20·37-s − 2·39-s + 8·43-s + 48-s + 3·49-s − 2·52-s − 8·57-s − 12·61-s + 2·63-s + 64-s − 16·67-s − 20·73-s − 6·75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 0.755·7-s + 1/3·9-s + 0.288·12-s − 0.554·13-s + 1/4·16-s − 1.83·19-s + 0.436·21-s − 6/5·25-s + 0.192·27-s + 0.377·28-s − 2.87·31-s + 1/6·36-s + 3.28·37-s − 0.320·39-s + 1.21·43-s + 0.144·48-s + 3/7·49-s − 0.277·52-s − 1.05·57-s − 1.53·61-s + 0.251·63-s + 1/8·64-s − 1.95·67-s − 2.34·73-s − 0.692·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 894348 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_1$ | \( 1 - T \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−7.927583890916278548349975461285, −7.53512261450424986470295233692, −7.31420502617006761020558985636, −6.70956799882972482046383355029, −6.02469114263822030525417968426, −5.87416482309586917103143441505, −5.34066639773140823665957604454, −4.51751350141054354627536433428, −4.08389192736738734823347258017, −4.05735746111134485794177861946, −2.81336235055681255128786870399, −2.70225474965229643722185344974, −1.85528951457630629269007018576, −1.52032901507690394890240915480, 0,
1.52032901507690394890240915480, 1.85528951457630629269007018576, 2.70225474965229643722185344974, 2.81336235055681255128786870399, 4.05735746111134485794177861946, 4.08389192736738734823347258017, 4.51751350141054354627536433428, 5.34066639773140823665957604454, 5.87416482309586917103143441505, 6.02469114263822030525417968426, 6.70956799882972482046383355029, 7.31420502617006761020558985636, 7.53512261450424986470295233692, 7.927583890916278548349975461285
