| L(s) = 1 | + 2·3-s + 3·9-s + 6·13-s − 4·17-s − 25-s + 4·27-s + 20·29-s + 12·39-s + 14·49-s − 8·51-s − 4·53-s + 4·61-s − 2·75-s − 32·79-s + 5·81-s + 40·87-s − 20·101-s − 8·103-s + 24·107-s + 12·113-s + 18·117-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 28·147-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 1.66·13-s − 0.970·17-s − 1/5·25-s + 0.769·27-s + 3.71·29-s + 1.92·39-s + 2·49-s − 1.12·51-s − 0.549·53-s + 0.512·61-s − 0.230·75-s − 3.60·79-s + 5/9·81-s + 4.28·87-s − 1.99·101-s − 0.788·103-s + 2.32·107-s + 1.12·113-s + 1.66·117-s + 1.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.30·147-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.461637546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.461637546\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 13 | $C_2$ | \( 1 - 6 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 174 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
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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
−8.627982756771823643577881755090, −8.619667257371961330067227491927, −8.285024864279606637065760013013, −8.029359623201156466436846610923, −7.18695031912853637754904230322, −7.15236980234186832278223578866, −6.75147762629991396435945289554, −6.23229570454298382183440511252, −5.93419797775499307442589579886, −5.59895186124173883871594295775, −4.68779540918092612991864887580, −4.60295192019775913744955954184, −4.19433077487200417405262122119, −3.75725378247589585663454287485, −3.06248938828455103746907107532, −3.01653119999157487364328290141, −2.38837273388352968302712019166, −1.87807108556414140793706343278, −1.20657724298935272356378157079, −0.75034358434948358531197603742,
0.75034358434948358531197603742, 1.20657724298935272356378157079, 1.87807108556414140793706343278, 2.38837273388352968302712019166, 3.01653119999157487364328290141, 3.06248938828455103746907107532, 3.75725378247589585663454287485, 4.19433077487200417405262122119, 4.60295192019775913744955954184, 4.68779540918092612991864887580, 5.59895186124173883871594295775, 5.93419797775499307442589579886, 6.23229570454298382183440511252, 6.75147762629991396435945289554, 7.15236980234186832278223578866, 7.18695031912853637754904230322, 8.029359623201156466436846610923, 8.285024864279606637065760013013, 8.619667257371961330067227491927, 8.627982756771823643577881755090
