| L(s) = 1 | + 2·3-s + 4-s − 3·9-s − 6·11-s + 2·12-s + 16-s + 6·23-s − 10·25-s − 14·27-s − 8·31-s − 12·33-s − 3·36-s + 4·37-s − 6·44-s + 2·48-s − 13·49-s − 6·53-s + 18·59-s + 64-s + 10·67-s + 12·69-s − 12·71-s − 20·75-s − 4·81-s − 24·89-s + 6·92-s − 16·93-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 9-s − 1.80·11-s + 0.577·12-s + 1/4·16-s + 1.25·23-s − 2·25-s − 2.69·27-s − 1.43·31-s − 2.08·33-s − 1/2·36-s + 0.657·37-s − 0.904·44-s + 0.288·48-s − 1.85·49-s − 0.824·53-s + 2.34·59-s + 1/8·64-s + 1.22·67-s + 1.44·69-s − 1.42·71-s − 2.30·75-s − 4/9·81-s − 2.54·89-s + 0.625·92-s − 1.65·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174724 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 11 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.684413720393242419277779815238, −8.560491774138482754724197116716, −7.88773793776252745940024787045, −7.73591843742988285704805588989, −7.25858039362668764589828470989, −6.52390816581107389509679557007, −5.81027585652647529822118508915, −5.50569298727080846748906445848, −5.08583557668970609096298687159, −4.11581948005602868281874533970, −3.39344369775766583219012744185, −2.99432139105097413830284496966, −2.42504146949326223577793697879, −1.89915892713560405275098298657, 0,
1.89915892713560405275098298657, 2.42504146949326223577793697879, 2.99432139105097413830284496966, 3.39344369775766583219012744185, 4.11581948005602868281874533970, 5.08583557668970609096298687159, 5.50569298727080846748906445848, 5.81027585652647529822118508915, 6.52390816581107389509679557007, 7.25858039362668764589828470989, 7.73591843742988285704805588989, 7.88773793776252745940024787045, 8.560491774138482754724197116716, 8.684413720393242419277779815238
