| L(s) = 1 | + 3-s + 8·7-s − 2·9-s − 8·13-s − 2·19-s + 8·21-s + 6·25-s − 5·27-s + 4·31-s − 8·39-s + 4·43-s + 35·49-s − 2·57-s − 16·63-s + 6·67-s + 14·73-s + 6·75-s + 81-s − 64·91-s + 4·93-s + 20·97-s + 28·103-s + 16·117-s + 2·121-s + 127-s + 4·129-s + 131-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 3.02·7-s − 2/3·9-s − 2.21·13-s − 0.458·19-s + 1.74·21-s + 6/5·25-s − 0.962·27-s + 0.718·31-s − 1.28·39-s + 0.609·43-s + 5·49-s − 0.264·57-s − 2.01·63-s + 0.733·67-s + 1.63·73-s + 0.692·75-s + 1/9·81-s − 6.70·91-s + 0.414·93-s + 2.03·97-s + 2.75·103-s + 1.47·117-s + 2/11·121-s + 0.0887·127-s + 0.352·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.151519339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.151519339\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 55 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 75 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 98 T^{2} + p^{2} T^{4} \) |
| 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
−8.144448745867564761668673879856, −7.85270807579471506940881517910, −7.54652659485024350146405875784, −7.14199575816468802585770550206, −6.53114999354080916433832754908, −5.84138364568425234720472786081, −5.18449816514671556410184821259, −5.01605790719167195114962081912, −4.67002564486747878008508820393, −4.24369875452449978793083688243, −3.42397017765064265977848133967, −2.58549028724135520777703456395, −2.28074214883869805118681726538, −1.81094489561324869795930126460, −0.833955193043309335124719700356,
0.833955193043309335124719700356, 1.81094489561324869795930126460, 2.28074214883869805118681726538, 2.58549028724135520777703456395, 3.42397017765064265977848133967, 4.24369875452449978793083688243, 4.67002564486747878008508820393, 5.01605790719167195114962081912, 5.18449816514671556410184821259, 5.84138364568425234720472786081, 6.53114999354080916433832754908, 7.14199575816468802585770550206, 7.54652659485024350146405875784, 7.85270807579471506940881517910, 8.144448745867564761668673879856
