L(s) = 1 | + 2·3-s + 5-s − 7-s + 9-s + 11-s + 2·13-s + 2·15-s + 6·17-s − 2·19-s − 2·21-s + 6·23-s + 25-s − 4·27-s − 8·31-s + 2·33-s − 35-s + 8·37-s + 4·39-s + 4·43-s + 45-s + 49-s + 12·51-s + 12·53-s + 55-s − 4·57-s − 12·59-s − 10·61-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s + 0.516·15-s + 1.45·17-s − 0.458·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s − 0.769·27-s − 1.43·31-s + 0.348·33-s − 0.169·35-s + 1.31·37-s + 0.640·39-s + 0.609·43-s + 0.149·45-s + 1/7·49-s + 1.68·51-s + 1.64·53-s + 0.134·55-s − 0.529·57-s − 1.56·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.543884109\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.543884109\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.099319795204037456385883035580, −7.49917720396822496122171266963, −6.73369281118297546350386344768, −5.89248012294307614707762919978, −5.32565853369495650763461508650, −4.17203829331023378906477864468, −3.45171834077435035734147444863, −2.87958199671111485592065236234, −1.98579761778179880510907193837, −0.967091792448194814804800261386,
0.967091792448194814804800261386, 1.98579761778179880510907193837, 2.87958199671111485592065236234, 3.45171834077435035734147444863, 4.17203829331023378906477864468, 5.32565853369495650763461508650, 5.89248012294307614707762919978, 6.73369281118297546350386344768, 7.49917720396822496122171266963, 8.099319795204037456385883035580