| L(s) = 1 | + 3-s − 4-s − 2·9-s − 3·11-s − 12-s + 16-s − 3·23-s − 5·27-s − 31-s − 3·33-s + 2·36-s − 3·37-s + 3·44-s − 3·47-s + 48-s − 5·49-s + 12·53-s + 12·59-s − 64-s + 6·67-s − 3·69-s − 6·71-s + 81-s + 21·89-s + 3·92-s − 93-s + 21·97-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1/2·4-s − 2/3·9-s − 0.904·11-s − 0.288·12-s + 1/4·16-s − 0.625·23-s − 0.962·27-s − 0.179·31-s − 0.522·33-s + 1/3·36-s − 0.493·37-s + 0.452·44-s − 0.437·47-s + 0.144·48-s − 5/7·49-s + 1.64·53-s + 1.56·59-s − 1/8·64-s + 0.733·67-s − 0.361·69-s − 0.712·71-s + 1/9·81-s + 2.22·89-s + 0.312·92-s − 0.103·93-s + 2.13·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2722500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_2$ | \( 1 + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 88 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
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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
−7.61803223569295674257357820068, −7.04144600753818729901446756765, −6.58089043479089203226227857957, −6.06226592711229873805457140919, −5.65066273488263285183621210379, −5.25815259123142587577135932819, −4.90183781136050579507498353677, −4.34113967317993007647843101464, −3.72209791622502394785449661096, −3.47672889803615794482068377787, −2.87763137195471371554477564941, −2.28658984805953971588113546637, −1.95017456688356022104862438035, −0.867025050825792368751523100866, 0,
0.867025050825792368751523100866, 1.95017456688356022104862438035, 2.28658984805953971588113546637, 2.87763137195471371554477564941, 3.47672889803615794482068377787, 3.72209791622502394785449661096, 4.34113967317993007647843101464, 4.90183781136050579507498353677, 5.25815259123142587577135932819, 5.65066273488263285183621210379, 6.06226592711229873805457140919, 6.58089043479089203226227857957, 7.04144600753818729901446756765, 7.61803223569295674257357820068
