| L(s) = 1 | + 3-s + 3·5-s − 2·9-s + 3·15-s + 3·23-s + 3·25-s − 5·27-s + 15·29-s + 7·43-s − 6·45-s − 9·47-s − 49-s + 12·53-s + 4·67-s + 3·69-s − 21·71-s + 3·73-s + 3·75-s + 81-s + 15·87-s + 26·97-s − 9·101-s + 9·115-s − 3·121-s + 6·125-s + 127-s + 7·129-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 2/3·9-s + 0.774·15-s + 0.625·23-s + 3/5·25-s − 0.962·27-s + 2.78·29-s + 1.06·43-s − 0.894·45-s − 1.31·47-s − 1/7·49-s + 1.64·53-s + 0.488·67-s + 0.361·69-s − 2.49·71-s + 0.351·73-s + 0.346·75-s + 1/9·81-s + 1.60·87-s + 2.63·97-s − 0.895·101-s + 0.839·115-s − 0.272·121-s + 0.536·125-s + 0.0887·127-s + 0.616·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.964741349\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.964741349\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−8.761037617717448726631057470312, −8.260605680190787776116556156846, −7.74589410062504641861770597076, −7.18215926579323538570077530593, −6.66624331913384797728712731348, −6.20803965176948035954556889231, −5.86060568650408566304659818762, −5.33131987034219782250326691061, −4.80572293368032746840732705810, −4.31559532653622868883890004391, −3.46728897439763695329170327740, −2.88508907144194422575730196846, −2.51974579098513689547785183389, −1.83935235418360773304272603797, −0.929795375200975086112929343726,
0.929795375200975086112929343726, 1.83935235418360773304272603797, 2.51974579098513689547785183389, 2.88508907144194422575730196846, 3.46728897439763695329170327740, 4.31559532653622868883890004391, 4.80572293368032746840732705810, 5.33131987034219782250326691061, 5.86060568650408566304659818762, 6.20803965176948035954556889231, 6.66624331913384797728712731348, 7.18215926579323538570077530593, 7.74589410062504641861770597076, 8.260605680190787776116556156846, 8.761037617717448726631057470312
