| L(s) = 1 | + 3·4-s + 4·7-s + 6·13-s + 5·16-s + 19-s + 8·25-s + 12·28-s − 10·37-s + 2·49-s + 18·52-s + 16·61-s + 3·64-s − 12·67-s + 4·73-s + 3·76-s − 12·79-s + 24·91-s + 6·97-s + 24·100-s − 4·103-s − 18·109-s + 20·112-s − 4·121-s + 127-s + 131-s + 4·133-s + 137-s + ⋯ |
| L(s) = 1 | + 3/2·4-s + 1.51·7-s + 1.66·13-s + 5/4·16-s + 0.229·19-s + 8/5·25-s + 2.26·28-s − 1.64·37-s + 2/7·49-s + 2.49·52-s + 2.04·61-s + 3/8·64-s − 1.46·67-s + 0.468·73-s + 0.344·76-s − 1.35·79-s + 2.51·91-s + 0.609·97-s + 12/5·100-s − 0.394·103-s − 1.72·109-s + 1.88·112-s − 0.363·121-s + 0.0887·127-s + 0.0873·131-s + 0.346·133-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 555579 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.139988395\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.139988395\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 116 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−8.408808558589246410485695702154, −7.988450749135605266304232794947, −7.55607400105650970541812714458, −7.04000217879605206023462372692, −6.66762904750664469138144687818, −6.31180265967519847762530496750, −5.69285112016661901477764413384, −5.22311882607786026931767703492, −4.84223630161015872072578480650, −4.05121634773197275874269817884, −3.55399156990710211213919138154, −2.90362156418189127855856742206, −2.31883092052134926433887065330, −1.45940222334492771658126789201, −1.30815728045188892153261341274,
1.30815728045188892153261341274, 1.45940222334492771658126789201, 2.31883092052134926433887065330, 2.90362156418189127855856742206, 3.55399156990710211213919138154, 4.05121634773197275874269817884, 4.84223630161015872072578480650, 5.22311882607786026931767703492, 5.69285112016661901477764413384, 6.31180265967519847762530496750, 6.66762904750664469138144687818, 7.04000217879605206023462372692, 7.55607400105650970541812714458, 7.988450749135605266304232794947, 8.408808558589246410485695702154
