L(s) = 1 | + 2·3-s + 3·9-s − 12·17-s + 4·19-s − 10·25-s + 4·27-s − 24·41-s − 8·43-s − 10·49-s − 24·51-s + 8·57-s + 24·59-s − 20·67-s + 28·73-s − 20·75-s + 5·81-s + 24·83-s − 20·97-s − 24·107-s + 12·113-s − 22·121-s − 48·123-s + 127-s − 16·129-s + 131-s + 137-s + 139-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 2.91·17-s + 0.917·19-s − 2·25-s + 0.769·27-s − 3.74·41-s − 1.21·43-s − 1.42·49-s − 3.36·51-s + 1.05·57-s + 3.12·59-s − 2.44·67-s + 3.27·73-s − 2.30·75-s + 5/9·81-s + 2.63·83-s − 2.03·97-s − 2.32·107-s + 1.12·113-s − 2·121-s − 4.32·123-s + 0.0887·127-s − 1.40·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.580237277156604970826791638060, −7.87419221261076033026145247629, −7.86871640882441783507560416572, −6.87432770376301100425942490139, −6.65843891177652373785895186115, −6.49041468304250674023097876601, −5.34555263887986009403036855109, −5.11454570698520102977754794044, −4.45279614679488683282940260914, −3.80067854107620357152276821608, −3.55171469212403510488492840860, −2.73602555300873188021493606065, −2.03881758362108754018941589683, −1.72702698258185110222331023684, 0,
1.72702698258185110222331023684, 2.03881758362108754018941589683, 2.73602555300873188021493606065, 3.55171469212403510488492840860, 3.80067854107620357152276821608, 4.45279614679488683282940260914, 5.11454570698520102977754794044, 5.34555263887986009403036855109, 6.49041468304250674023097876601, 6.65843891177652373785895186115, 6.87432770376301100425942490139, 7.86871640882441783507560416572, 7.87419221261076033026145247629, 8.580237277156604970826791638060