L(s) = 1 | − 2.39·2-s + 3-s + 3.74·4-s − 0.714·5-s − 2.39·6-s − 0.642·7-s − 4.17·8-s + 9-s + 1.71·10-s − 11-s + 3.74·12-s + 3.57·13-s + 1.53·14-s − 0.714·15-s + 2.51·16-s − 0.970·17-s − 2.39·18-s − 1.66·19-s − 2.67·20-s − 0.642·21-s + 2.39·22-s − 6.84·23-s − 4.17·24-s − 4.48·25-s − 8.57·26-s + 27-s − 2.40·28-s + ⋯ |
L(s) = 1 | − 1.69·2-s + 0.577·3-s + 1.87·4-s − 0.319·5-s − 0.978·6-s − 0.242·7-s − 1.47·8-s + 0.333·9-s + 0.541·10-s − 0.301·11-s + 1.07·12-s + 0.992·13-s + 0.411·14-s − 0.184·15-s + 0.628·16-s − 0.235·17-s − 0.564·18-s − 0.382·19-s − 0.597·20-s − 0.140·21-s + 0.510·22-s − 1.42·23-s − 0.851·24-s − 0.897·25-s − 1.68·26-s + 0.192·27-s − 0.454·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 5 | \( 1 + 0.714T + 5T^{2} \) |
| 7 | \( 1 + 0.642T + 7T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 + 0.970T + 17T^{2} \) |
| 19 | \( 1 + 1.66T + 19T^{2} \) |
| 23 | \( 1 + 6.84T + 23T^{2} \) |
| 29 | \( 1 - 2.83T + 29T^{2} \) |
| 31 | \( 1 - 4.42T + 31T^{2} \) |
| 37 | \( 1 - 1.60T + 37T^{2} \) |
| 41 | \( 1 + 7.68T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 9.96T + 47T^{2} \) |
| 53 | \( 1 + 4.23T + 53T^{2} \) |
| 59 | \( 1 - 2.72T + 59T^{2} \) |
| 67 | \( 1 + 15.4T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 2.89T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 - 3.90T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 6.96T + 97T^{2} \) |
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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
−8.723632121734576256603179144222, −8.041518518160920928554944642074, −7.72972756122648176847077466587, −6.60302262887237385998444973664, −6.07657047195582261024333930022, −4.50895510622470733888886594889, −3.48558746444345653542264905337, −2.40179782589570265557673097331, −1.43751455783935622869623836519, 0,
1.43751455783935622869623836519, 2.40179782589570265557673097331, 3.48558746444345653542264905337, 4.50895510622470733888886594889, 6.07657047195582261024333930022, 6.60302262887237385998444973664, 7.72972756122648176847077466587, 8.041518518160920928554944642074, 8.723632121734576256603179144222