| L(s) = 1 | − 2-s − 2·3-s − 4-s − 4·5-s + 2·6-s + 3·8-s + 3·9-s + 4·10-s + 2·12-s + 8·15-s − 16-s − 12·17-s − 3·18-s + 19-s + 4·20-s − 6·24-s + 2·25-s − 4·27-s − 8·30-s − 16·31-s − 5·32-s + 12·34-s − 3·36-s − 38-s − 12·40-s − 12·45-s + 2·48-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s − 1.78·5-s + 0.816·6-s + 1.06·8-s + 9-s + 1.26·10-s + 0.577·12-s + 2.06·15-s − 1/4·16-s − 2.91·17-s − 0.707·18-s + 0.229·19-s + 0.894·20-s − 1.22·24-s + 2/5·25-s − 0.769·27-s − 1.46·30-s − 2.87·31-s − 0.883·32-s + 2.05·34-s − 1/2·36-s − 0.162·38-s − 1.89·40-s − 1.78·45-s + 0.288·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 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
−8.099644677533565214444467559710, −7.43259308735181576062150293394, −6.98236656892345640095319736227, −6.88281746236137995334549719629, −6.30774150521524867405980638808, −5.42974656732833448500123753190, −5.28029214515602867786662657712, −4.62864849822679992384179631378, −4.16838172205224943400021226417, −3.91841567788800315953565193679, −3.49609663069962101813276746331, −2.25179019930198933577153857080, −1.72203721066548968493435337820, −0.49711700208122723907813480784, 0,
0.49711700208122723907813480784, 1.72203721066548968493435337820, 2.25179019930198933577153857080, 3.49609663069962101813276746331, 3.91841567788800315953565193679, 4.16838172205224943400021226417, 4.62864849822679992384179631378, 5.28029214515602867786662657712, 5.42974656732833448500123753190, 6.30774150521524867405980638808, 6.88281746236137995334549719629, 6.98236656892345640095319736227, 7.43259308735181576062150293394, 8.099644677533565214444467559710
