| L(s) = 1 | − 4·3-s + 6·9-s + 6·11-s − 6·17-s − 2·19-s − 7·25-s + 4·27-s − 24·33-s − 12·41-s + 2·43-s − 11·49-s + 24·51-s + 8·57-s + 12·59-s − 8·67-s + 2·73-s + 28·75-s − 37·81-s + 24·83-s + 8·97-s + 36·99-s + 12·107-s − 36·113-s + 5·121-s + 48·123-s + 127-s − 8·129-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s + 1.80·11-s − 1.45·17-s − 0.458·19-s − 7/5·25-s + 0.769·27-s − 4.17·33-s − 1.87·41-s + 0.304·43-s − 1.57·49-s + 3.36·51-s + 1.05·57-s + 1.56·59-s − 0.977·67-s + 0.234·73-s + 3.23·75-s − 4.11·81-s + 2.63·83-s + 0.812·97-s + 3.61·99-s + 1.16·107-s − 3.38·113-s + 5/11·121-s + 4.32·123-s + 0.0887·127-s − 0.704·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23658496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 130 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
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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
−7.890098194081383287401025279219, −7.86247178170391644231281725966, −6.96483992214302621772782299976, −6.74773789847706140614702122488, −6.61087264466814400114919753128, −6.29948891499780074984969109326, −6.03271744589960930199647233250, −5.51194389006493875070341859208, −5.27643413595791840557371315705, −4.85345080123550281169158570368, −4.36238618832976049279431756856, −4.28662839023784457144388060225, −3.50608547955028473788542284958, −3.43073510329859146904243713501, −2.42275743228573031372196572663, −2.07675659219047051748061649060, −1.39940359562148139047100422170, −0.982578941467437418203549684328, 0, 0,
0.982578941467437418203549684328, 1.39940359562148139047100422170, 2.07675659219047051748061649060, 2.42275743228573031372196572663, 3.43073510329859146904243713501, 3.50608547955028473788542284958, 4.28662839023784457144388060225, 4.36238618832976049279431756856, 4.85345080123550281169158570368, 5.27643413595791840557371315705, 5.51194389006493875070341859208, 6.03271744589960930199647233250, 6.29948891499780074984969109326, 6.61087264466814400114919753128, 6.74773789847706140614702122488, 6.96483992214302621772782299976, 7.86247178170391644231281725966, 7.890098194081383287401025279219
