| L(s) = 1 | − 4-s + 2·7-s − 10·13-s − 3·16-s − 2·19-s − 2·28-s + 10·31-s + 2·37-s + 2·43-s − 11·49-s + 10·52-s + 4·61-s + 7·64-s − 16·67-s − 4·73-s + 2·76-s − 2·79-s − 20·91-s − 34·97-s − 16·103-s + 34·109-s − 6·112-s − 10·121-s − 10·124-s + 127-s + 131-s − 4·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 0.755·7-s − 2.77·13-s − 3/4·16-s − 0.458·19-s − 0.377·28-s + 1.79·31-s + 0.328·37-s + 0.304·43-s − 1.57·49-s + 1.38·52-s + 0.512·61-s + 7/8·64-s − 1.95·67-s − 0.468·73-s + 0.229·76-s − 0.225·79-s − 2.09·91-s − 3.45·97-s − 1.57·103-s + 3.25·109-s − 0.566·112-s − 0.909·121-s − 0.898·124-s + 0.0887·127-s + 0.0873·131-s − 0.346·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36905625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36905625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 106 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 118 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 17 T + p T^{2} )^{2} \) |
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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
−7.73113150341185910820842660006, −7.63951183173300067446920335433, −7.35820863115049951375983169541, −6.67262370490231006217841087430, −6.64751101991616912042654855359, −6.26477624188867309715648966133, −5.42952938180779524455698201945, −5.42645229258182234815286880670, −4.93224326096937188053609094765, −4.59268889916187232916769130579, −4.34051215050290813326163622817, −4.17554336273533188573557838776, −3.35607553124950933923278322217, −2.81281709134313959588118517057, −2.57464864174709277275780744543, −2.21764780567849678940369277775, −1.60890610249545408237814534707, −1.05463291020775159793841319825, 0, 0,
1.05463291020775159793841319825, 1.60890610249545408237814534707, 2.21764780567849678940369277775, 2.57464864174709277275780744543, 2.81281709134313959588118517057, 3.35607553124950933923278322217, 4.17554336273533188573557838776, 4.34051215050290813326163622817, 4.59268889916187232916769130579, 4.93224326096937188053609094765, 5.42645229258182234815286880670, 5.42952938180779524455698201945, 6.26477624188867309715648966133, 6.64751101991616912042654855359, 6.67262370490231006217841087430, 7.35820863115049951375983169541, 7.63951183173300067446920335433, 7.73113150341185910820842660006
