L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 3·8-s + 9-s + 10-s − 12-s + 15-s − 16-s + 18-s − 6·19-s − 20-s − 12·23-s − 3·24-s + 25-s + 27-s − 10·29-s + 30-s + 5·32-s − 36-s − 6·38-s − 3·40-s − 10·43-s + 45-s − 12·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 0.258·15-s − 1/4·16-s + 0.235·18-s − 1.37·19-s − 0.223·20-s − 2.50·23-s − 0.612·24-s + 1/5·25-s + 0.192·27-s − 1.85·29-s + 0.182·30-s + 0.883·32-s − 1/6·36-s − 0.973·38-s − 0.474·40-s − 1.52·43-s + 0.149·45-s − 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$ | \( 1 - T \) |
good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 24 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.841949077573873823775079160332, −8.310387580710565750282513871774, −8.008759299199905597127093432791, −7.45120708943997382225768151918, −6.67801778768693653434556586990, −6.35074992708740092614652664848, −5.66703120125243249571427201763, −5.49998553895872731415445036150, −4.64979802762485321002813298462, −4.04402494779224559503968925162, −3.89398238296327538196455237767, −3.09438497492793793598861161837, −2.27164495510320738928114371029, −1.76895553779176625944886599601, 0,
1.76895553779176625944886599601, 2.27164495510320738928114371029, 3.09438497492793793598861161837, 3.89398238296327538196455237767, 4.04402494779224559503968925162, 4.64979802762485321002813298462, 5.49998553895872731415445036150, 5.66703120125243249571427201763, 6.35074992708740092614652664848, 6.67801778768693653434556586990, 7.45120708943997382225768151918, 8.008759299199905597127093432791, 8.310387580710565750282513871774, 8.841949077573873823775079160332