| L(s) = 1 | + 4-s + 6·5-s − 3·11-s + 16-s + 6·20-s − 12·23-s + 17·25-s + 10·31-s + 4·37-s − 3·44-s + 12·47-s − 13·49-s + 18·53-s − 18·55-s + 24·59-s + 64-s + 28·67-s + 6·80-s − 36·89-s − 12·92-s − 2·97-s + 17·100-s − 8·103-s − 12·113-s − 72·115-s − 2·121-s + 10·124-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 2.68·5-s − 0.904·11-s + 1/4·16-s + 1.34·20-s − 2.50·23-s + 17/5·25-s + 1.79·31-s + 0.657·37-s − 0.452·44-s + 1.75·47-s − 1.85·49-s + 2.47·53-s − 2.42·55-s + 3.12·59-s + 1/8·64-s + 3.42·67-s + 0.670·80-s − 3.81·89-s − 1.25·92-s − 0.203·97-s + 1.69·100-s − 0.788·103-s − 1.12·113-s − 6.71·115-s − 0.181·121-s + 0.898·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352836 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352836 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.398114398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.398114398\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−8.628140728035937440023020847412, −8.268048909391662018895990050852, −8.015255053006877536003075086198, −7.12468139474762615848955209063, −6.73860900674537002760989556093, −6.29574961437633563574123917046, −5.84315760589664083898483470884, −5.39074241880952165115421872247, −5.37363015537505503550836847886, −4.30456169908575776721292993281, −3.80538107085536458846182311961, −2.58138575780524151090389938489, −2.44850978333263328730546301919, −2.03246862806699813641002498927, −1.09285120951784850918046733268,
1.09285120951784850918046733268, 2.03246862806699813641002498927, 2.44850978333263328730546301919, 2.58138575780524151090389938489, 3.80538107085536458846182311961, 4.30456169908575776721292993281, 5.37363015537505503550836847886, 5.39074241880952165115421872247, 5.84315760589664083898483470884, 6.29574961437633563574123917046, 6.73860900674537002760989556093, 7.12468139474762615848955209063, 8.015255053006877536003075086198, 8.268048909391662018895990050852, 8.628140728035937440023020847412
