| L(s) = 1 | + 2-s + 3-s + 4-s − 3·5-s + 6-s + 8-s + 9-s − 3·10-s + 12-s − 3·15-s + 16-s + 18-s − 3·20-s − 12·23-s + 24-s + 8·25-s + 27-s − 29-s − 3·30-s + 32-s + 36-s − 3·40-s − 2·43-s − 3·45-s − 12·46-s − 4·47-s + 48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.34·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.948·10-s + 0.288·12-s − 0.774·15-s + 1/4·16-s + 0.235·18-s − 0.670·20-s − 2.50·23-s + 0.204·24-s + 8/5·25-s + 0.192·27-s − 0.185·29-s − 0.547·30-s + 0.176·32-s + 1/6·36-s − 0.474·40-s − 0.304·43-s − 0.447·45-s − 1.76·46-s − 0.583·47-s + 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 501120 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
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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
−8.207534854290089911295485219373, −7.75134068311223529148940269890, −7.51530735559548585460618458743, −6.99633989267962531454267266827, −6.44440270644592242066530792317, −5.98529039617744244472665046687, −5.47385165656180121102572930508, −4.69079125766459332667344208567, −4.40628766862947527093679277236, −3.87995385293391431431761298284, −3.49440028353102624158950603925, −2.89836228771972304330528256877, −2.22278419197095632149218406217, −1.42239085163224965424579119906, 0,
1.42239085163224965424579119906, 2.22278419197095632149218406217, 2.89836228771972304330528256877, 3.49440028353102624158950603925, 3.87995385293391431431761298284, 4.40628766862947527093679277236, 4.69079125766459332667344208567, 5.47385165656180121102572930508, 5.98529039617744244472665046687, 6.44440270644592242066530792317, 6.99633989267962531454267266827, 7.51530735559548585460618458743, 7.75134068311223529148940269890, 8.207534854290089911295485219373
