L(s) = 1 | − 2.79·2-s − 3-s + 5.81·4-s + 0.438·5-s + 2.79·6-s − 7-s − 10.6·8-s + 9-s − 1.22·10-s − 5.11·11-s − 5.81·12-s + 3.57·13-s + 2.79·14-s − 0.438·15-s + 18.1·16-s − 3.22·17-s − 2.79·18-s + 4.20·19-s + 2.55·20-s + 21-s + 14.2·22-s + 5.74·23-s + 10.6·24-s − 4.80·25-s − 9.99·26-s − 27-s − 5.81·28-s + ⋯ |
L(s) = 1 | − 1.97·2-s − 0.577·3-s + 2.90·4-s + 0.196·5-s + 1.14·6-s − 0.377·7-s − 3.76·8-s + 0.333·9-s − 0.387·10-s − 1.54·11-s − 1.67·12-s + 0.991·13-s + 0.747·14-s − 0.113·15-s + 4.54·16-s − 0.781·17-s − 0.658·18-s + 0.963·19-s + 0.570·20-s + 0.218·21-s + 3.04·22-s + 1.19·23-s + 2.17·24-s − 0.961·25-s − 1.96·26-s − 0.192·27-s − 1.09·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4364349922\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4364349922\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
good | 2 | \( 1 + 2.79T + 2T^{2} \) |
| 5 | \( 1 - 0.438T + 5T^{2} \) |
| 11 | \( 1 + 5.11T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 + 3.22T + 17T^{2} \) |
| 19 | \( 1 - 4.20T + 19T^{2} \) |
| 23 | \( 1 - 5.74T + 23T^{2} \) |
| 29 | \( 1 - 4.80T + 29T^{2} \) |
| 31 | \( 1 - 4.33T + 31T^{2} \) |
| 37 | \( 1 + 1.07T + 37T^{2} \) |
| 41 | \( 1 + 4.91T + 41T^{2} \) |
| 43 | \( 1 + 9.33T + 43T^{2} \) |
| 47 | \( 1 + 4.27T + 47T^{2} \) |
| 53 | \( 1 - 2.74T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 + 2.23T + 61T^{2} \) |
| 67 | \( 1 + 2.87T + 67T^{2} \) |
| 71 | \( 1 + 8.53T + 71T^{2} \) |
| 73 | \( 1 + 1.54T + 73T^{2} \) |
| 79 | \( 1 + 7.00T + 79T^{2} \) |
| 83 | \( 1 - 6.26T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 - 9.32T + 97T^{2} \) |
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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
−8.458614487285124101340086211752, −7.916337128900924367455948182699, −7.08743636213313421800270341686, −6.54854670474666309003473462461, −5.81555849795872581261752835821, −5.03311726474272061748010154202, −3.35416888994388295523485629607, −2.62260899418188059302887036238, −1.56678247114431785699539528275, −0.53058928115590208043935819731,
0.53058928115590208043935819731, 1.56678247114431785699539528275, 2.62260899418188059302887036238, 3.35416888994388295523485629607, 5.03311726474272061748010154202, 5.81555849795872581261752835821, 6.54854670474666309003473462461, 7.08743636213313421800270341686, 7.916337128900924367455948182699, 8.458614487285124101340086211752