| L(s) = 1 | − 3-s + 3·5-s − 2·9-s − 3·15-s − 4·19-s − 3·23-s − 25-s + 5·27-s − 3·29-s + 11·43-s − 6·45-s + 3·47-s + 49-s + 4·57-s + 8·67-s + 3·69-s − 27·71-s − 23·73-s + 75-s + 81-s + 3·87-s − 12·95-s − 2·97-s + 15·101-s − 9·115-s + 5·121-s − 18·125-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s − 2/3·9-s − 0.774·15-s − 0.917·19-s − 0.625·23-s − 1/5·25-s + 0.962·27-s − 0.557·29-s + 1.67·43-s − 0.894·45-s + 0.437·47-s + 1/7·49-s + 0.529·57-s + 0.977·67-s + 0.361·69-s − 3.20·71-s − 2.69·73-s + 0.115·75-s + 1/9·81-s + 0.321·87-s − 1.23·95-s − 0.203·97-s + 1.49·101-s − 0.839·115-s + 5/11·121-s − 1.60·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 52 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 71 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 + 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 173 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
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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
−8.557024956556078308625831646377, −7.75545169554458583650269754075, −7.50077507358292180474752055094, −6.86170028461518629205900682659, −6.21983200993035242053454943614, −5.95959601166052613923811284842, −5.74641610741098821814352300551, −5.23605605021411542879308293182, −4.48347441546354958099512101674, −4.13593813452022316454104369131, −3.29378274328924768143290995434, −2.54856853592737338313737706947, −2.12917892162220754584503832706, −1.32772972033511259901790240734, 0,
1.32772972033511259901790240734, 2.12917892162220754584503832706, 2.54856853592737338313737706947, 3.29378274328924768143290995434, 4.13593813452022316454104369131, 4.48347441546354958099512101674, 5.23605605021411542879308293182, 5.74641610741098821814352300551, 5.95959601166052613923811284842, 6.21983200993035242053454943614, 6.86170028461518629205900682659, 7.50077507358292180474752055094, 7.75545169554458583650269754075, 8.557024956556078308625831646377
