L(s) = 1 | − 1.17·2-s − 3-s − 0.609·4-s + 4.22·5-s + 1.17·6-s + 3.50·7-s + 3.07·8-s + 9-s − 4.98·10-s + 1.99·11-s + 0.609·12-s − 13-s − 4.12·14-s − 4.22·15-s − 2.40·16-s − 1.98·17-s − 1.17·18-s − 0.207·19-s − 2.57·20-s − 3.50·21-s − 2.34·22-s − 2.99·23-s − 3.07·24-s + 12.8·25-s + 1.17·26-s − 27-s − 2.13·28-s + ⋯ |
L(s) = 1 | − 0.833·2-s − 0.577·3-s − 0.304·4-s + 1.89·5-s + 0.481·6-s + 1.32·7-s + 1.08·8-s + 0.333·9-s − 1.57·10-s + 0.600·11-s + 0.176·12-s − 0.277·13-s − 1.10·14-s − 1.09·15-s − 0.602·16-s − 0.482·17-s − 0.277·18-s − 0.0476·19-s − 0.576·20-s − 0.764·21-s − 0.500·22-s − 0.625·23-s − 0.628·24-s + 2.57·25-s + 0.231·26-s − 0.192·27-s − 0.403·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.676242643\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.676242643\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 + 1.17T + 2T^{2} \) |
| 5 | \( 1 - 4.22T + 5T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 11 | \( 1 - 1.99T + 11T^{2} \) |
| 17 | \( 1 + 1.98T + 17T^{2} \) |
| 19 | \( 1 + 0.207T + 19T^{2} \) |
| 23 | \( 1 + 2.99T + 23T^{2} \) |
| 29 | \( 1 - 6.92T + 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 - 0.697T + 37T^{2} \) |
| 41 | \( 1 + 3.09T + 41T^{2} \) |
| 43 | \( 1 - 8.08T + 43T^{2} \) |
| 47 | \( 1 - 6.61T + 47T^{2} \) |
| 53 | \( 1 + 4.57T + 53T^{2} \) |
| 59 | \( 1 - 9.27T + 59T^{2} \) |
| 61 | \( 1 + 4.61T + 61T^{2} \) |
| 67 | \( 1 - 5.99T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 8.59T + 73T^{2} \) |
| 79 | \( 1 - 6.17T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 + 8.17T + 89T^{2} \) |
| 97 | \( 1 + 0.583T + 97T^{2} \) |
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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
−8.625982741210425093749239368615, −7.88004384432480343951819204759, −6.95333956620187785649481854203, −6.25089977288188278629221454822, −5.43081050636726631350185256854, −4.87289193206079356646073248200, −4.17264563736865992748295560939, −2.41239452597431535036347986819, −1.68176853559829857833124870661, −0.964712498468719322404545947494,
0.964712498468719322404545947494, 1.68176853559829857833124870661, 2.41239452597431535036347986819, 4.17264563736865992748295560939, 4.87289193206079356646073248200, 5.43081050636726631350185256854, 6.25089977288188278629221454822, 6.95333956620187785649481854203, 7.88004384432480343951819204759, 8.625982741210425093749239368615