| L(s) = 1 | + 4-s − 3·7-s − 2·13-s + 16-s + 19-s − 8·25-s − 3·28-s + 6·31-s − 6·37-s − 11·43-s − 2·52-s − 61-s + 64-s − 12·67-s + 10·73-s + 76-s − 22·79-s + 6·91-s + 3·97-s − 8·100-s + 30·103-s − 28·109-s − 3·112-s + 7·121-s + 6·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 1.13·7-s − 0.554·13-s + 1/4·16-s + 0.229·19-s − 8/5·25-s − 0.566·28-s + 1.07·31-s − 0.986·37-s − 1.67·43-s − 0.277·52-s − 0.128·61-s + 1/8·64-s − 1.46·67-s + 1.17·73-s + 0.114·76-s − 2.47·79-s + 0.628·91-s + 0.304·97-s − 4/5·100-s + 2.95·103-s − 2.68·109-s − 0.283·112-s + 7/11·121-s + 0.538·124-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165564 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 9 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 51 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
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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
−8.996294878625520744685378376293, −8.509491327817171022160684827427, −7.970330301213339827373042607252, −7.45651823186292180514994218249, −7.03927173318081883993435711235, −6.45118743752527956902700378369, −6.17272617959686255877189418263, −5.54797574939072751523807856865, −4.96753901853904199068994176696, −4.28707851185001509248160586494, −3.55154322352609679636083965926, −3.11006437759544936704912910044, −2.40108066866787107831530510295, −1.55726954268813622718957120242, 0,
1.55726954268813622718957120242, 2.40108066866787107831530510295, 3.11006437759544936704912910044, 3.55154322352609679636083965926, 4.28707851185001509248160586494, 4.96753901853904199068994176696, 5.54797574939072751523807856865, 6.17272617959686255877189418263, 6.45118743752527956902700378369, 7.03927173318081883993435711235, 7.45651823186292180514994218249, 7.970330301213339827373042607252, 8.509491327817171022160684827427, 8.996294878625520744685378376293
