| L(s) = 1 | − 3-s − 16·7-s − 8·9-s − 4·13-s + 22·19-s + 16·21-s + 17·27-s − 92·31-s + 32·37-s + 4·39-s − 124·43-s + 94·49-s − 22·57-s − 32·61-s + 128·63-s − 226·67-s − 202·73-s + 136·79-s + 55·81-s + 64·91-s + 92·93-s + 44·97-s − 52·103-s + 352·109-s − 32·111-s + 32·117-s − 73·121-s + ⋯ |
| L(s) = 1 | − 1/3·3-s − 2.28·7-s − 8/9·9-s − 0.307·13-s + 1.15·19-s + 0.761·21-s + 0.629·27-s − 2.96·31-s + 0.864·37-s + 4/39·39-s − 2.88·43-s + 1.91·49-s − 0.385·57-s − 0.524·61-s + 2.03·63-s − 3.37·67-s − 2.76·73-s + 1.72·79-s + 0.679·81-s + 0.703·91-s + 0.989·93-s + 0.453·97-s − 0.504·103-s + 3.22·109-s − 0.288·111-s + 0.273·117-s − 0.603·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2591145110\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2591145110\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 29 T + p^{2} T^{2} )( 1 + 29 T + p^{2} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 202 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 422 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 16 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 527 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 62 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 3158 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4358 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1922 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 113 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 1258 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 101 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 68 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 13463 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 13007 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2} \) |
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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
−12.15765111793053226969972882269, −11.32314994118341667389154363753, −10.84661852940972404532680115808, −10.19739360360986183982715067682, −9.916747465624891889209550840271, −9.458390124849823287851288606533, −8.938785858761975445950802467361, −8.737652391024185121131660011102, −7.72153956458941178226485922323, −7.34841724765203637668411687462, −6.86860326362973904427192882929, −6.14507178889110193025756837554, −6.01549815096420120498686756195, −5.34552455857369841628635571314, −4.75754471503577270226583644173, −3.73806419816889662133776980963, −3.13388297968388083664314454966, −3.03150199046996107002501097680, −1.74875850009925863695828967600, −0.24284099450680643829129442202,
0.24284099450680643829129442202, 1.74875850009925863695828967600, 3.03150199046996107002501097680, 3.13388297968388083664314454966, 3.73806419816889662133776980963, 4.75754471503577270226583644173, 5.34552455857369841628635571314, 6.01549815096420120498686756195, 6.14507178889110193025756837554, 6.86860326362973904427192882929, 7.34841724765203637668411687462, 7.72153956458941178226485922323, 8.737652391024185121131660011102, 8.938785858761975445950802467361, 9.458390124849823287851288606533, 9.916747465624891889209550840271, 10.19739360360986183982715067682, 10.84661852940972404532680115808, 11.32314994118341667389154363753, 12.15765111793053226969972882269
