| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s − 4·13-s − 2·15-s + 12·17-s − 8·19-s − 8·21-s + 25-s − 4·27-s + 12·29-s + 4·35-s − 4·37-s − 8·39-s − 45-s − 2·49-s + 24·51-s − 16·57-s − 4·63-s + 4·65-s − 24·71-s + 2·75-s − 11·81-s − 12·83-s − 12·85-s + 24·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.10·13-s − 0.516·15-s + 2.91·17-s − 1.83·19-s − 1.74·21-s + 1/5·25-s − 0.769·27-s + 2.22·29-s + 0.676·35-s − 0.657·37-s − 1.28·39-s − 0.149·45-s − 2/7·49-s + 3.36·51-s − 2.11·57-s − 0.503·63-s + 0.496·65-s − 2.84·71-s + 0.230·75-s − 1.22·81-s − 1.31·83-s − 1.30·85-s + 2.57·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.414042470938792927705837581638, −8.354217905819492276622407670543, −7.68981058172178643196515579808, −7.38409000222510611898226618037, −6.79665463493597297389359578264, −6.30016858748825461633606551620, −5.85740317846722582719715899176, −5.17381357315305902780091527317, −4.55074829093045740786740684957, −3.90074996505307731038192172560, −3.35135398071564319960087881565, −2.91285425746382377562417869193, −2.58321256178540655780606724063, −1.41105370921345955286948063209, 0,
1.41105370921345955286948063209, 2.58321256178540655780606724063, 2.91285425746382377562417869193, 3.35135398071564319960087881565, 3.90074996505307731038192172560, 4.55074829093045740786740684957, 5.17381357315305902780091527317, 5.85740317846722582719715899176, 6.30016858748825461633606551620, 6.79665463493597297389359578264, 7.38409000222510611898226618037, 7.68981058172178643196515579808, 8.354217905819492276622407670543, 8.414042470938792927705837581638
