L(s) = 1 | − 2-s − 4-s + 3·8-s − 4·13-s − 16-s − 6·17-s − 8·19-s − 25-s + 4·26-s − 5·32-s + 6·34-s + 8·38-s + 2·43-s + 18·47-s − 8·49-s + 50-s + 4·52-s − 8·53-s − 16·59-s + 7·64-s + 6·67-s + 6·68-s + 8·76-s − 9·81-s − 26·83-s − 2·86-s − 18·94-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.10·13-s − 1/4·16-s − 1.45·17-s − 1.83·19-s − 1/5·25-s + 0.784·26-s − 0.883·32-s + 1.02·34-s + 1.29·38-s + 0.304·43-s + 2.62·47-s − 8/7·49-s + 0.141·50-s + 0.554·52-s − 1.09·53-s − 2.08·59-s + 7/8·64-s + 0.733·67-s + 0.727·68-s + 0.917·76-s − 81-s − 2.85·83-s − 0.215·86-s − 1.85·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 17 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 120 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
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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
−10.21297280232623962707754863672, −9.708254981814360858292663699128, −9.202072821355533222558179266565, −8.662434285870896733895612797810, −8.423005787233979230346840942154, −7.53639490843311349121052775111, −7.27812164456876842814157434085, −6.46471330296411122639951342724, −5.93303168624518243897643367682, −4.94139277905990953926192582146, −4.48404698138439520044486747643, −3.99612885944062491515261866857, −2.68559110089798692482543213228, −1.85990320101749540748192083329, 0,
1.85990320101749540748192083329, 2.68559110089798692482543213228, 3.99612885944062491515261866857, 4.48404698138439520044486747643, 4.94139277905990953926192582146, 5.93303168624518243897643367682, 6.46471330296411122639951342724, 7.27812164456876842814157434085, 7.53639490843311349121052775111, 8.423005787233979230346840942154, 8.662434285870896733895612797810, 9.202072821355533222558179266565, 9.708254981814360858292663699128, 10.21297280232623962707754863672