L(s) = 1 | + 12.2·3-s + 9.62e3·7-s − 1.95e4·9-s − 5.56e4·11-s − 1.69e5·13-s − 2.07e5·17-s − 8.02e5·19-s + 1.17e5·21-s − 1.24e6·23-s − 4.78e5·27-s − 4.28e6·29-s + 3.58e6·31-s − 6.79e5·33-s + 2.89e6·37-s − 2.07e6·39-s + 2.51e7·41-s − 2.00e7·43-s + 3.73e7·47-s + 5.22e7·49-s − 2.53e6·51-s + 2.55e7·53-s − 9.79e6·57-s + 9.96e7·59-s + 2.00e8·61-s − 1.87e8·63-s − 8.09e7·67-s − 1.51e7·69-s + ⋯ |
L(s) = 1 | + 0.0870·3-s + 1.51·7-s − 0.992·9-s − 1.14·11-s − 1.64·13-s − 0.602·17-s − 1.41·19-s + 0.131·21-s − 0.925·23-s − 0.173·27-s − 1.12·29-s + 0.697·31-s − 0.0997·33-s + 0.254·37-s − 0.143·39-s + 1.39·41-s − 0.893·43-s + 1.11·47-s + 1.29·49-s − 0.0524·51-s + 0.444·53-s − 0.122·57-s + 1.07·59-s + 1.85·61-s − 1.50·63-s − 0.491·67-s − 0.0805·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.174714189\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.174714189\) |
\(L(\frac{11}{2})\) |
|
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 | \( 1 - 12.2T + 1.96e4T^{2} \) |
| 7 | \( 1 - 9.62e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.56e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.69e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.07e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 8.02e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.24e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.28e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 3.58e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 2.89e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 2.51e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.00e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.73e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 2.55e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.96e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 2.00e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 8.09e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 4.31e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.40e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.81e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 6.01e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.39e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.23e8T + 7.60e17T^{2} \) |
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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
−9.824699251864237703822195011254, −8.575573685387857101593383689217, −8.057751739306022178482963632140, −7.20878395773451171769088877321, −5.78140983981250689512392324572, −5.01962688825234605250198325793, −4.18656834028466691642015723888, −2.43541760848382070913734855595, −2.16195920886305474303606885900, −0.41762187556954241902528467500,
0.41762187556954241902528467500, 2.16195920886305474303606885900, 2.43541760848382070913734855595, 4.18656834028466691642015723888, 5.01962688825234605250198325793, 5.78140983981250689512392324572, 7.20878395773451171769088877321, 8.057751739306022178482963632140, 8.575573685387857101593383689217, 9.824699251864237703822195011254