L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 9-s + 2·10-s + 13-s + 16-s − 8·17-s + 18-s + 2·20-s − 25-s + 26-s − 14·29-s + 32-s − 8·34-s + 36-s − 14·37-s + 2·40-s + 8·41-s + 2·45-s + 9·49-s − 50-s + 52-s − 4·53-s − 14·58-s − 4·61-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.277·13-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 0.447·20-s − 1/5·25-s + 0.196·26-s − 2.59·29-s + 0.176·32-s − 1.37·34-s + 1/6·36-s − 2.30·37-s + 0.316·40-s + 1.24·41-s + 0.298·45-s + 9/7·49-s − 0.141·50-s + 0.138·52-s − 0.549·53-s − 1.83·58-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 - T + p T^{2} \) |
good | 7 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 45 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 41 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 13 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 65 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 119 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
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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
−7.65621015624670339223977324278, −7.16748559967744894773946903357, −6.91134745448266764176535778726, −6.48075238670997440009746086916, −5.84937041357114864062577702167, −5.68764517015168307043462131790, −5.19614134629359471045037754767, −4.64309860475878813669982821617, −4.01790493677731378029390436025, −3.86811752572588622453463978000, −3.09605780546570593206962775316, −2.36432308198299391440034354298, −1.97719593813255056338628051399, −1.47495497295792512433291695321, 0,
1.47495497295792512433291695321, 1.97719593813255056338628051399, 2.36432308198299391440034354298, 3.09605780546570593206962775316, 3.86811752572588622453463978000, 4.01790493677731378029390436025, 4.64309860475878813669982821617, 5.19614134629359471045037754767, 5.68764517015168307043462131790, 5.84937041357114864062577702167, 6.48075238670997440009746086916, 6.91134745448266764176535778726, 7.16748559967744894773946903357, 7.65621015624670339223977324278