L(s) = 1 | − 0.820·2-s + 3-s − 1.32·4-s + 3.24·5-s − 0.820·6-s − 1.69·7-s + 2.72·8-s + 9-s − 2.66·10-s − 2.47·11-s − 1.32·12-s − 13-s + 1.39·14-s + 3.24·15-s + 0.413·16-s − 0.439·17-s − 0.820·18-s − 2.19·19-s − 4.31·20-s − 1.69·21-s + 2.02·22-s + 0.232·23-s + 2.72·24-s + 5.56·25-s + 0.820·26-s + 27-s + 2.25·28-s + ⋯ |
L(s) = 1 | − 0.580·2-s + 0.577·3-s − 0.663·4-s + 1.45·5-s − 0.334·6-s − 0.641·7-s + 0.965·8-s + 0.333·9-s − 0.843·10-s − 0.745·11-s − 0.382·12-s − 0.277·13-s + 0.372·14-s + 0.839·15-s + 0.103·16-s − 0.106·17-s − 0.193·18-s − 0.503·19-s − 0.964·20-s − 0.370·21-s + 0.432·22-s + 0.0485·23-s + 0.557·24-s + 1.11·25-s + 0.160·26-s + 0.192·27-s + 0.425·28-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.820T + 2T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 7 | \( 1 + 1.69T + 7T^{2} \) |
| 11 | \( 1 + 2.47T + 11T^{2} \) |
| 17 | \( 1 + 0.439T + 17T^{2} \) |
| 19 | \( 1 + 2.19T + 19T^{2} \) |
| 23 | \( 1 - 0.232T + 23T^{2} \) |
| 29 | \( 1 + 3.67T + 29T^{2} \) |
| 31 | \( 1 + 7.98T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 - 2.73T + 41T^{2} \) |
| 43 | \( 1 + 6.45T + 43T^{2} \) |
| 47 | \( 1 - 4.09T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 8.89T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 - 2.29T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 - 3.40T + 73T^{2} \) |
| 79 | \( 1 - 1.40T + 79T^{2} \) |
| 83 | \( 1 + 6.57T + 83T^{2} \) |
| 89 | \( 1 - 9.38T + 89T^{2} \) |
| 97 | \( 1 + 14.6T + 97T^{2} \) |
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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
−8.185980040180968042872139593129, −7.55021285454094690162091493312, −6.67503649274720224558160246803, −5.82322730576249040805715574420, −5.16991155733682109049716116759, −4.28521110284215527474389191569, −3.23959877975180723445633320122, −2.30836396127493035413230815038, −1.51229357085470165655104196778, 0,
1.51229357085470165655104196778, 2.30836396127493035413230815038, 3.23959877975180723445633320122, 4.28521110284215527474389191569, 5.16991155733682109049716116759, 5.82322730576249040805715574420, 6.67503649274720224558160246803, 7.55021285454094690162091493312, 8.185980040180968042872139593129