| L(s) = 1 | − 2·3-s + 3·9-s + 8·11-s − 4·17-s − 8·19-s − 10·25-s − 4·27-s − 16·33-s + 12·41-s + 8·43-s + 2·49-s + 8·51-s + 16·57-s − 24·59-s + 24·67-s − 12·73-s + 20·75-s + 5·81-s − 8·83-s − 12·89-s − 4·97-s + 24·99-s + 8·107-s − 28·113-s + 26·121-s − 24·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s + 2.41·11-s − 0.970·17-s − 1.83·19-s − 2·25-s − 0.769·27-s − 2.78·33-s + 1.87·41-s + 1.21·43-s + 2/7·49-s + 1.12·51-s + 2.11·57-s − 3.12·59-s + 2.93·67-s − 1.40·73-s + 2.30·75-s + 5/9·81-s − 0.878·83-s − 1.27·89-s − 0.406·97-s + 2.41·99-s + 0.773·107-s − 2.63·113-s + 2.36·121-s − 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.247152815550235840332214408381, −7.67021342089506565111723692624, −6.99687341654975900890511495598, −6.85175430383558681973846914652, −6.23593170660239061738441922167, −5.88446326475989803483764829308, −5.83902316255569103541752486301, −4.70084538751769099501127599916, −4.33363432781740332926422055062, −4.09550440201686426997590498271, −3.62331160644145024955043318233, −2.46196604328535614137231641027, −1.85805342505545187947666684739, −1.15979350949225944879282791156, 0,
1.15979350949225944879282791156, 1.85805342505545187947666684739, 2.46196604328535614137231641027, 3.62331160644145024955043318233, 4.09550440201686426997590498271, 4.33363432781740332926422055062, 4.70084538751769099501127599916, 5.83902316255569103541752486301, 5.88446326475989803483764829308, 6.23593170660239061738441922167, 6.85175430383558681973846914652, 6.99687341654975900890511495598, 7.67021342089506565111723692624, 8.247152815550235840332214408381
