| L(s) = 1 | + 4-s + 9-s − 3·16-s − 2·17-s − 4·23-s + 3·25-s − 14·29-s + 36-s − 12·43-s + 6·49-s − 22·53-s + 2·61-s − 7·64-s − 2·68-s − 8·79-s + 81-s − 4·92-s + 3·100-s + 2·101-s + 12·103-s + 12·107-s − 10·113-s − 14·116-s − 2·121-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1/3·9-s − 3/4·16-s − 0.485·17-s − 0.834·23-s + 3/5·25-s − 2.59·29-s + 1/6·36-s − 1.82·43-s + 6/7·49-s − 3.02·53-s + 0.256·61-s − 7/8·64-s − 0.242·68-s − 0.900·79-s + 1/9·81-s − 0.417·92-s + 3/10·100-s + 0.199·101-s + 1.18·103-s + 1.16·107-s − 0.940·113-s − 1.29·116-s − 0.181·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257049 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 53 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 9 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 49 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 110 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.650001915547516127708968168795, −8.254312306204594442663740138509, −7.64830789971015885999624459773, −7.24284512977827471313557365200, −6.89695511683785395962493269837, −6.19987347920647555245778860339, −6.02542306843297483164581786835, −5.17000017954860320631534903426, −4.79061112061585039281209559118, −4.11886089601568935163628455261, −3.56967166346421752643069142460, −2.91773453960743377742523935696, −2.03598784293238748245786017561, −1.65401027593800212162746376412, 0,
1.65401027593800212162746376412, 2.03598784293238748245786017561, 2.91773453960743377742523935696, 3.56967166346421752643069142460, 4.11886089601568935163628455261, 4.79061112061585039281209559118, 5.17000017954860320631534903426, 6.02542306843297483164581786835, 6.19987347920647555245778860339, 6.89695511683785395962493269837, 7.24284512977827471313557365200, 7.64830789971015885999624459773, 8.254312306204594442663740138509, 8.650001915547516127708968168795
