| L(s) = 1 | − 2·4-s − 2·5-s − 2·11-s + 2·13-s − 6·19-s + 4·20-s + 10·23-s − 7·25-s + 16·29-s − 22·41-s − 18·43-s + 4·44-s − 12·49-s − 4·52-s + 4·55-s + 8·64-s − 4·65-s − 20·67-s + 20·71-s + 12·76-s − 20·92-s + 12·95-s + 14·100-s + 10·103-s − 14·107-s + 10·113-s − 20·115-s + ⋯ |
| L(s) = 1 | − 4-s − 0.894·5-s − 0.603·11-s + 0.554·13-s − 1.37·19-s + 0.894·20-s + 2.08·23-s − 7/5·25-s + 2.97·29-s − 3.43·41-s − 2.74·43-s + 0.603·44-s − 1.71·49-s − 0.554·52-s + 0.539·55-s + 64-s − 0.496·65-s − 2.44·67-s + 2.37·71-s + 1.37·76-s − 2.08·92-s + 1.23·95-s + 7/5·100-s + 0.985·103-s − 1.35·107-s + 0.940·113-s − 1.86·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6765201 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6765201 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 17 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 12 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 108 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
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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
−8.457082831516044211034578429793, −8.360100864596840843858824042230, −8.150476895713820543385545537081, −7.79691265281051227930607866587, −7.01228030851359434126049935377, −6.72622308619066112764533449728, −6.55493244990709712735737303754, −6.10984218965192148580390478757, −5.31187253318571871459919454951, −4.92698454103035525172474583499, −4.81038919621767539359377199819, −4.50013119829340008410672241173, −3.65698459889650032998969921090, −3.61368485568681109035597776554, −3.05128961370685963844137340381, −2.51344594506643183186678679784, −1.72016940680890540653911574939, −1.17327626079928707214046596014, 0, 0,
1.17327626079928707214046596014, 1.72016940680890540653911574939, 2.51344594506643183186678679784, 3.05128961370685963844137340381, 3.61368485568681109035597776554, 3.65698459889650032998969921090, 4.50013119829340008410672241173, 4.81038919621767539359377199819, 4.92698454103035525172474583499, 5.31187253318571871459919454951, 6.10984218965192148580390478757, 6.55493244990709712735737303754, 6.72622308619066112764533449728, 7.01228030851359434126049935377, 7.79691265281051227930607866587, 8.150476895713820543385545537081, 8.360100864596840843858824042230, 8.457082831516044211034578429793
