| L(s) = 1 | + 4-s − 5·9-s − 6·11-s + 16-s + 10·19-s + 4·31-s − 5·36-s − 6·41-s − 6·44-s − 10·49-s + 4·61-s + 64-s + 24·71-s + 10·76-s − 20·79-s + 16·81-s + 30·89-s + 30·99-s − 36·101-s − 20·109-s + 5·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s − 5·144-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 5/3·9-s − 1.80·11-s + 1/4·16-s + 2.29·19-s + 0.718·31-s − 5/6·36-s − 0.937·41-s − 0.904·44-s − 1.42·49-s + 0.512·61-s + 1/8·64-s + 2.84·71-s + 1.14·76-s − 2.25·79-s + 16/9·81-s + 3.17·89-s + 3.01·99-s − 3.58·101-s − 1.91·109-s + 5/11·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.416·144-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6823642606\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6823642606\) |
| \(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−13.31818220322005518851751496924, −12.18462032318833367073561378768, −11.89949732310571442955739052562, −11.16007422368021302024475397710, −10.81431448769284119152476376034, −9.930723730317650982781533521356, −9.447991916742081591691197868932, −8.316812877075225116243141130520, −8.071729218562685355881418106440, −7.29853134793764088505009214313, −6.35306147256574072432179047412, −5.34945783064988058695168525193, −5.20592148788576143134616479765, −3.29699975126484092788987476482, −2.65155040417235216240951299295,
2.65155040417235216240951299295, 3.29699975126484092788987476482, 5.20592148788576143134616479765, 5.34945783064988058695168525193, 6.35306147256574072432179047412, 7.29853134793764088505009214313, 8.071729218562685355881418106440, 8.316812877075225116243141130520, 9.447991916742081591691197868932, 9.930723730317650982781533521356, 10.81431448769284119152476376034, 11.16007422368021302024475397710, 11.89949732310571442955739052562, 12.18462032318833367073561378768, 13.31818220322005518851751496924
