| L(s) = 1 | − 2·3-s + 7-s + 3·9-s + 8·19-s − 2·21-s − 6·25-s − 4·27-s + 12·29-s + 16·31-s − 20·37-s + 16·47-s + 49-s + 12·53-s − 16·57-s − 8·59-s + 3·63-s + 12·75-s + 5·81-s − 8·83-s − 24·87-s − 32·93-s + 28·109-s + 40·111-s − 28·113-s − 22·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 9-s + 1.83·19-s − 0.436·21-s − 6/5·25-s − 0.769·27-s + 2.22·29-s + 2.87·31-s − 3.28·37-s + 2.33·47-s + 1/7·49-s + 1.64·53-s − 2.11·57-s − 1.04·59-s + 0.377·63-s + 1.38·75-s + 5/9·81-s − 0.878·83-s − 2.57·87-s − 3.31·93-s + 2.68·109-s + 3.79·111-s − 2.63·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.396240030\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.396240030\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.840136214566098327187778551790, −8.122692960827942227844556545585, −7.65690854393686671838881168913, −7.25581286887766067123055482424, −6.63994290005683855125700712975, −6.46532614984807246340960494164, −5.57316575173703563161974438677, −5.53186445958724562402483080296, −4.86424457503876839761931685454, −4.45555499882348184354185481660, −3.85201838137013588199771793801, −3.09950638537541493859717484704, −2.46249542885781697802628622502, −1.41812868870774364880044650320, −0.78183674791366035764093383220,
0.78183674791366035764093383220, 1.41812868870774364880044650320, 2.46249542885781697802628622502, 3.09950638537541493859717484704, 3.85201838137013588199771793801, 4.45555499882348184354185481660, 4.86424457503876839761931685454, 5.53186445958724562402483080296, 5.57316575173703563161974438677, 6.46532614984807246340960494164, 6.63994290005683855125700712975, 7.25581286887766067123055482424, 7.65690854393686671838881168913, 8.122692960827942227844556545585, 8.840136214566098327187778551790
