| L(s) = 1 | − 2·4-s − 4·5-s + 9-s + 6·13-s + 4·16-s − 6·17-s + 8·20-s + 2·25-s − 2·36-s + 16·37-s − 14·41-s − 4·45-s − 10·49-s − 12·52-s + 6·53-s − 16·61-s − 8·64-s − 24·65-s + 12·68-s + 24·73-s − 16·80-s + 81-s + 24·85-s + 20·89-s + 22·97-s − 4·100-s − 18·101-s + ⋯ |
| L(s) = 1 | − 4-s − 1.78·5-s + 1/3·9-s + 1.66·13-s + 16-s − 1.45·17-s + 1.78·20-s + 2/5·25-s − 1/3·36-s + 2.63·37-s − 2.18·41-s − 0.596·45-s − 1.42·49-s − 1.66·52-s + 0.824·53-s − 2.04·61-s − 64-s − 2.97·65-s + 1.45·68-s + 2.80·73-s − 1.78·80-s + 1/9·81-s + 2.60·85-s + 2.11·89-s + 2.23·97-s − 2/5·100-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
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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
−8.510537086333408831410494114245, −8.311202822356102145576413186265, −7.80436893404074773763481968895, −7.51672561268763905855633948060, −6.78461901069866544425570526686, −6.19603689777594065734989014157, −5.96338923144874103909303632728, −4.79535808046119236916099761001, −4.74387622427760402111791858890, −4.10319001261157529164243453614, −3.50887355939377684370334646650, −3.47776319336265678852062283368, −2.16208352631894123622226457901, −1.04596298856260690087985554195, 0,
1.04596298856260690087985554195, 2.16208352631894123622226457901, 3.47776319336265678852062283368, 3.50887355939377684370334646650, 4.10319001261157529164243453614, 4.74387622427760402111791858890, 4.79535808046119236916099761001, 5.96338923144874103909303632728, 6.19603689777594065734989014157, 6.78461901069866544425570526686, 7.51672561268763905855633948060, 7.80436893404074773763481968895, 8.311202822356102145576413186265, 8.510537086333408831410494114245
