| L(s) = 1 | − 6·7-s − 2·13-s − 4·16-s + 2·19-s + 2·25-s − 4·31-s + 10·37-s + 8·43-s + 17·49-s + 10·61-s − 10·67-s + 10·73-s + 14·79-s + 12·91-s + 18·97-s + 18·103-s − 12·109-s + 24·112-s − 10·121-s + 127-s + 131-s − 12·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 2.26·7-s − 0.554·13-s − 16-s + 0.458·19-s + 2/5·25-s − 0.718·31-s + 1.64·37-s + 1.21·43-s + 17/7·49-s + 1.28·61-s − 1.22·67-s + 1.17·73-s + 1.57·79-s + 1.25·91-s + 1.82·97-s + 1.77·103-s − 1.14·109-s + 2.26·112-s − 0.909·121-s + 0.0887·127-s + 0.0873·131-s − 1.04·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 31 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
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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
−7.974728759160587023789168250825, −7.52511142952145981163200404744, −7.21657895026143250951482792776, −6.63849658081134314972172268796, −6.35176589928687028911514119730, −6.02316516967709113242582870957, −5.37743998111742888238457626329, −4.86191666079255482292079990676, −4.26219951834931360138424798653, −3.65838792827931529908210393063, −3.31070845176509611431043865672, −2.53421397007909328606022910301, −2.34006644478051300829209147005, −0.934836773566159816026389665737, 0,
0.934836773566159816026389665737, 2.34006644478051300829209147005, 2.53421397007909328606022910301, 3.31070845176509611431043865672, 3.65838792827931529908210393063, 4.26219951834931360138424798653, 4.86191666079255482292079990676, 5.37743998111742888238457626329, 6.02316516967709113242582870957, 6.35176589928687028911514119730, 6.63849658081134314972172268796, 7.21657895026143250951482792776, 7.52511142952145981163200404744, 7.974728759160587023789168250825
