| L(s) = 1 | − 3-s − 2·7-s − 2·9-s + 2·13-s − 12·19-s + 2·21-s + 25-s + 5·27-s + 12·37-s − 2·39-s + 12·43-s + 3·49-s + 12·57-s − 8·61-s + 4·63-s − 8·67-s + 20·73-s − 75-s − 6·79-s + 81-s − 4·91-s + 14·97-s − 18·103-s − 22·109-s − 12·111-s − 4·117-s + 3·121-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s + 0.554·13-s − 2.75·19-s + 0.436·21-s + 1/5·25-s + 0.962·27-s + 1.97·37-s − 0.320·39-s + 1.82·43-s + 3/7·49-s + 1.58·57-s − 1.02·61-s + 0.503·63-s − 0.977·67-s + 2.34·73-s − 0.115·75-s − 0.675·79-s + 1/9·81-s − 0.419·91-s + 1.42·97-s − 1.77·103-s − 2.10·109-s − 1.13·111-s − 0.369·117-s + 3/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
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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
−8.321037332106296729315848288758, −7.61068895852562797070369378134, −7.16216561796569421885219307291, −6.45877795748859351665331711034, −6.17106315121314565062659046983, −6.14206553924090701620723770176, −5.45734437575009036836059962399, −4.83171016164985859832688933282, −4.17514659999582833871651190557, −4.06182673812785905054132006247, −3.16093778472040229481013774454, −2.60467600555756229149551481858, −2.11101541760584806136071858440, −0.949618884734753047450408637076, 0,
0.949618884734753047450408637076, 2.11101541760584806136071858440, 2.60467600555756229149551481858, 3.16093778472040229481013774454, 4.06182673812785905054132006247, 4.17514659999582833871651190557, 4.83171016164985859832688933282, 5.45734437575009036836059962399, 6.14206553924090701620723770176, 6.17106315121314565062659046983, 6.45877795748859351665331711034, 7.16216561796569421885219307291, 7.61068895852562797070369378134, 8.321037332106296729315848288758
