| L(s) = 1 | − 2·2-s + 2·4-s − 3·9-s − 4·16-s + 6·18-s + 8·25-s − 14·29-s + 8·32-s − 6·36-s − 16·50-s + 10·53-s + 28·58-s − 8·64-s + 9·81-s + 16·100-s − 20·106-s − 2·113-s − 28·116-s + 127-s + 131-s + 137-s + 139-s + 12·144-s + 149-s + 151-s + 157-s − 18·162-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 9-s − 16-s + 1.41·18-s + 8/5·25-s − 2.59·29-s + 1.41·32-s − 36-s − 2.26·50-s + 1.37·53-s + 3.67·58-s − 64-s + 81-s + 8/5·100-s − 1.94·106-s − 0.188·113-s − 2.59·116-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s − 1.41·162-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 16 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
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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
−8.675283624900062511911811378656, −8.165842182931156078591588403838, −7.56253085249857986653649458051, −7.43401045105870059031051173050, −6.73681136581834265637925026910, −6.39296396190320632860668721558, −5.59206924634252637463300387111, −5.35483700669784098380769784887, −4.64926701107503784437807501566, −3.94848150715579718577789265617, −3.32917196730094299212080403177, −2.57865934666678914372372469635, −2.00275142514673925542798845843, −1.07413156948196004775276684427, 0,
1.07413156948196004775276684427, 2.00275142514673925542798845843, 2.57865934666678914372372469635, 3.32917196730094299212080403177, 3.94848150715579718577789265617, 4.64926701107503784437807501566, 5.35483700669784098380769784887, 5.59206924634252637463300387111, 6.39296396190320632860668721558, 6.73681136581834265637925026910, 7.43401045105870059031051173050, 7.56253085249857986653649458051, 8.165842182931156078591588403838, 8.675283624900062511911811378656
