| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s + 4·6-s − 2·7-s − 3·9-s − 4·12-s + 4·14-s − 4·16-s + 6·18-s + 4·21-s − 9·25-s + 14·27-s − 4·28-s + 14·31-s + 8·32-s − 6·36-s + 6·37-s − 8·42-s + 16·47-s + 8·48-s − 3·49-s + 18·50-s − 12·53-s − 28·54-s + 10·59-s − 28·62-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s + 1.63·6-s − 0.755·7-s − 9-s − 1.15·12-s + 1.06·14-s − 16-s + 1.41·18-s + 0.872·21-s − 9/5·25-s + 2.69·27-s − 0.755·28-s + 2.51·31-s + 1.41·32-s − 36-s + 0.986·37-s − 1.23·42-s + 2.33·47-s + 1.15·48-s − 3/7·49-s + 2.54·50-s − 1.64·53-s − 3.81·54-s + 1.30·59-s − 3.55·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.490415131274113451268545814290, −8.719810569642884578389491013532, −8.603539619290756001226038948684, −7.87989145823484455720674220186, −7.59753326202792059542705479631, −6.69785642750182140490053057916, −6.36261389471308870138602900888, −5.96253635220307759773371824019, −5.41592840817145656472425824771, −4.67360648409355241573689732264, −4.01118939616306336398601307633, −2.92516350065322057627024861621, −2.40668113522951484990555404309, −0.979033992832471368219893964268, 0,
0.979033992832471368219893964268, 2.40668113522951484990555404309, 2.92516350065322057627024861621, 4.01118939616306336398601307633, 4.67360648409355241573689732264, 5.41592840817145656472425824771, 5.96253635220307759773371824019, 6.36261389471308870138602900888, 6.69785642750182140490053057916, 7.59753326202792059542705479631, 7.87989145823484455720674220186, 8.603539619290756001226038948684, 8.719810569642884578389491013532, 9.490415131274113451268545814290
