L(s) = 1 | − 0.732·3-s + 1.26·7-s − 2.46·9-s − 3.46·11-s + 3.46·13-s − 3.46·17-s + 2·19-s − 0.928·21-s + 8.19·23-s + 4·27-s − 9.46·31-s + 2.53·33-s + 6·37-s − 2.53·39-s − 2.53·41-s − 10.1·43-s + 8.19·47-s − 5.39·49-s + 2.53·51-s − 10.3·53-s − 1.46·57-s + 6·59-s + 12.9·61-s − 3.12·63-s + 10.1·67-s − 6·69-s + 4.39·71-s + ⋯ |
L(s) = 1 | − 0.422·3-s + 0.479·7-s − 0.821·9-s − 1.04·11-s + 0.960·13-s − 0.840·17-s + 0.458·19-s − 0.202·21-s + 1.70·23-s + 0.769·27-s − 1.69·31-s + 0.441·33-s + 0.986·37-s − 0.406·39-s − 0.396·41-s − 1.55·43-s + 1.19·47-s − 0.770·49-s + 0.355·51-s − 1.42·53-s − 0.193·57-s + 0.781·59-s + 1.65·61-s − 0.393·63-s + 1.24·67-s − 0.722·69-s + 0.521·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.356819852\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.356819852\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 0.732T + 3T^{2} \) |
| 7 | \( 1 - 1.26T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 8.19T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 9.46T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 10.1T + 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−8.144658587975875353275521982021, −7.25931454523435601040537904145, −6.63488319512698082331812558322, −5.73144761451614474294172144249, −5.26172614377723480287251438802, −4.62323425191528013398175988362, −3.50961479417966787121207993830, −2.81559230919673045102680642592, −1.80367652396824163872344005306, −0.61137964985745817595391094137,
0.61137964985745817595391094137, 1.80367652396824163872344005306, 2.81559230919673045102680642592, 3.50961479417966787121207993830, 4.62323425191528013398175988362, 5.26172614377723480287251438802, 5.73144761451614474294172144249, 6.63488319512698082331812558322, 7.25931454523435601040537904145, 8.144658587975875353275521982021