L(s) = 1 | − 0.139·2-s − 3-s − 1.98·4-s − 5-s + 0.139·6-s − 3.46·7-s + 0.554·8-s + 9-s + 0.139·10-s + 1.14·11-s + 1.98·12-s + 2.00·13-s + 0.482·14-s + 15-s + 3.88·16-s + 0.449·17-s − 0.139·18-s − 7.45·19-s + 1.98·20-s + 3.46·21-s − 0.158·22-s − 3.71·23-s − 0.554·24-s + 25-s − 0.279·26-s − 27-s + 6.86·28-s + ⋯ |
L(s) = 1 | − 0.0984·2-s − 0.577·3-s − 0.990·4-s − 0.447·5-s + 0.0568·6-s − 1.31·7-s + 0.195·8-s + 0.333·9-s + 0.0440·10-s + 0.343·11-s + 0.571·12-s + 0.556·13-s + 0.128·14-s + 0.258·15-s + 0.971·16-s + 0.108·17-s − 0.0328·18-s − 1.71·19-s + 0.442·20-s + 0.756·21-s − 0.0338·22-s − 0.773·23-s − 0.113·24-s + 0.200·25-s − 0.0547·26-s − 0.192·27-s + 1.29·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 + 0.139T + 2T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 1.14T + 11T^{2} \) |
| 13 | \( 1 - 2.00T + 13T^{2} \) |
| 17 | \( 1 - 0.449T + 17T^{2} \) |
| 19 | \( 1 + 7.45T + 19T^{2} \) |
| 23 | \( 1 + 3.71T + 23T^{2} \) |
| 29 | \( 1 + 7.06T + 29T^{2} \) |
| 31 | \( 1 - 9.74T + 31T^{2} \) |
| 37 | \( 1 + 9.44T + 37T^{2} \) |
| 41 | \( 1 - 5.26T + 41T^{2} \) |
| 43 | \( 1 - 10.1T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 - 2.48T + 53T^{2} \) |
| 59 | \( 1 - 3.76T + 59T^{2} \) |
| 61 | \( 1 - 10.6T + 61T^{2} \) |
| 67 | \( 1 + 2.31T + 67T^{2} \) |
| 71 | \( 1 - 2.65T + 71T^{2} \) |
| 73 | \( 1 + 9.32T + 73T^{2} \) |
| 79 | \( 1 + 4.78T + 79T^{2} \) |
| 83 | \( 1 + 3.53T + 83T^{2} \) |
| 89 | \( 1 - 12.1T + 89T^{2} \) |
| 97 | \( 1 - 9.30T + 97T^{2} \) |
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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
−7.71217420083701889470009669175, −6.97367565091776035759178231689, −6.10339085210299798774528412563, −5.83324169257116192698984968120, −4.68261912079806720504678204805, −3.97752758739851604660727633732, −3.61085557887953356345300874978, −2.33222193857631381013681448898, −0.882997523421309787544009117131, 0,
0.882997523421309787544009117131, 2.33222193857631381013681448898, 3.61085557887953356345300874978, 3.97752758739851604660727633732, 4.68261912079806720504678204805, 5.83324169257116192698984968120, 6.10339085210299798774528412563, 6.97367565091776035759178231689, 7.71217420083701889470009669175