| L(s) = 1 | + 3-s + 4-s + 4·7-s − 2·9-s + 12-s − 8·13-s + 16-s + 10·19-s + 4·21-s − 5·27-s + 4·28-s + 4·31-s − 2·36-s + 4·37-s − 8·39-s − 8·43-s + 48-s − 2·49-s − 8·52-s + 10·57-s + 4·61-s − 8·63-s + 64-s − 26·67-s + 22·73-s + 10·76-s − 20·79-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/2·4-s + 1.51·7-s − 2/3·9-s + 0.288·12-s − 2.21·13-s + 1/4·16-s + 2.29·19-s + 0.872·21-s − 0.962·27-s + 0.755·28-s + 0.718·31-s − 1/3·36-s + 0.657·37-s − 1.28·39-s − 1.21·43-s + 0.144·48-s − 2/7·49-s − 1.10·52-s + 1.32·57-s + 0.512·61-s − 1.00·63-s + 1/8·64-s − 3.17·67-s + 2.57·73-s + 1.14·76-s − 2.25·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.644486109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.644486109\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−10.81431448769284119152476376034, −10.18789957492033670124802464882, −9.481094579683641415186193216160, −9.447991916742081591691197868932, −8.316812877075225116243141130520, −8.114692805267148178564014319106, −7.40487753845237746133032109213, −7.29853134793764088505009214313, −6.28477478517999960844504626470, −5.23380204015925208976616676091, −5.20592148788576143134616479765, −4.36763710614752510700235050554, −3.13165277251593234405465211013, −2.65155040417235216240951299295, −1.63862417385241407780156087735,
1.63862417385241407780156087735, 2.65155040417235216240951299295, 3.13165277251593234405465211013, 4.36763710614752510700235050554, 5.20592148788576143134616479765, 5.23380204015925208976616676091, 6.28477478517999960844504626470, 7.29853134793764088505009214313, 7.40487753845237746133032109213, 8.114692805267148178564014319106, 8.316812877075225116243141130520, 9.447991916742081591691197868932, 9.481094579683641415186193216160, 10.18789957492033670124802464882, 10.81431448769284119152476376034
