L(s) = 1 | + 2-s − 1.69·3-s + 4-s − 5-s − 1.69·6-s + 4.41·7-s + 8-s − 0.116·9-s − 10-s − 5.34·11-s − 1.69·12-s − 13-s + 4.41·14-s + 1.69·15-s + 16-s − 5.12·17-s − 0.116·18-s + 7.74·19-s − 20-s − 7.50·21-s − 5.34·22-s + 3.66·23-s − 1.69·24-s + 25-s − 26-s + 5.29·27-s + 4.41·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.980·3-s + 0.5·4-s − 0.447·5-s − 0.693·6-s + 1.66·7-s + 0.353·8-s − 0.0389·9-s − 0.316·10-s − 1.61·11-s − 0.490·12-s − 0.277·13-s + 1.18·14-s + 0.438·15-s + 0.250·16-s − 1.24·17-s − 0.0275·18-s + 1.77·19-s − 0.223·20-s − 1.63·21-s − 1.14·22-s + 0.763·23-s − 0.346·24-s + 0.200·25-s − 0.196·26-s + 1.01·27-s + 0.834·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 + 1.69T + 3T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 + 5.34T + 11T^{2} \) |
| 17 | \( 1 + 5.12T + 17T^{2} \) |
| 19 | \( 1 - 7.74T + 19T^{2} \) |
| 23 | \( 1 - 3.66T + 23T^{2} \) |
| 29 | \( 1 + 3.96T + 29T^{2} \) |
| 37 | \( 1 + 3.45T + 37T^{2} \) |
| 41 | \( 1 + 11.5T + 41T^{2} \) |
| 43 | \( 1 + 6.03T + 43T^{2} \) |
| 47 | \( 1 + 9.45T + 47T^{2} \) |
| 53 | \( 1 + 7.79T + 53T^{2} \) |
| 59 | \( 1 - 2.68T + 59T^{2} \) |
| 61 | \( 1 + 7.65T + 61T^{2} \) |
| 67 | \( 1 - 9.27T + 67T^{2} \) |
| 71 | \( 1 - 4.68T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 9.54T + 83T^{2} \) |
| 89 | \( 1 - 7.99T + 89T^{2} \) |
| 97 | \( 1 + 14.2T + 97T^{2} \) |
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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
−8.017023369369741739410910059465, −7.26159991374525007145095254831, −6.57297267453723327494662001023, −5.36261759357826903578372969930, −5.06949362808233965041024306346, −4.80655438206054988739131673381, −3.48401599782395734335930531003, −2.53902005028705099244245624358, −1.46041542421377256089717123979, 0,
1.46041542421377256089717123979, 2.53902005028705099244245624358, 3.48401599782395734335930531003, 4.80655438206054988739131673381, 5.06949362808233965041024306346, 5.36261759357826903578372969930, 6.57297267453723327494662001023, 7.26159991374525007145095254831, 8.017023369369741739410910059465