L(s) = 1 | − 2.54·2-s + 3-s + 4.48·4-s + 3.86·5-s − 2.54·6-s − 7-s − 6.32·8-s + 9-s − 9.84·10-s + 0.530·11-s + 4.48·12-s − 3.89·13-s + 2.54·14-s + 3.86·15-s + 7.13·16-s − 0.207·17-s − 2.54·18-s − 1.98·19-s + 17.3·20-s − 21-s − 1.35·22-s − 6.59·23-s − 6.32·24-s + 9.94·25-s + 9.90·26-s + 27-s − 4.48·28-s + ⋯ |
L(s) = 1 | − 1.80·2-s + 0.577·3-s + 2.24·4-s + 1.72·5-s − 1.03·6-s − 0.377·7-s − 2.23·8-s + 0.333·9-s − 3.11·10-s + 0.159·11-s + 1.29·12-s − 1.07·13-s + 0.680·14-s + 0.998·15-s + 1.78·16-s − 0.0503·17-s − 0.600·18-s − 0.454·19-s + 3.87·20-s − 0.218·21-s − 0.288·22-s − 1.37·23-s − 1.29·24-s + 1.98·25-s + 1.94·26-s + 0.192·27-s − 0.847·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 + 2.54T + 2T^{2} \) |
| 5 | \( 1 - 3.86T + 5T^{2} \) |
| 11 | \( 1 - 0.530T + 11T^{2} \) |
| 13 | \( 1 + 3.89T + 13T^{2} \) |
| 17 | \( 1 + 0.207T + 17T^{2} \) |
| 19 | \( 1 + 1.98T + 19T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 - 7.18T + 29T^{2} \) |
| 31 | \( 1 + 7.37T + 31T^{2} \) |
| 37 | \( 1 + 9.83T + 37T^{2} \) |
| 41 | \( 1 + 6.61T + 41T^{2} \) |
| 43 | \( 1 - 7.11T + 43T^{2} \) |
| 47 | \( 1 + 4.71T + 47T^{2} \) |
| 53 | \( 1 + 9.30T + 53T^{2} \) |
| 59 | \( 1 - 9.81T + 59T^{2} \) |
| 61 | \( 1 - 4.17T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 4.15T + 71T^{2} \) |
| 73 | \( 1 - 8.57T + 73T^{2} \) |
| 79 | \( 1 - 13.3T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 4.45T + 89T^{2} \) |
| 97 | \( 1 + 1.62T + 97T^{2} \) |
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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
−7.67823462040903217127011323866, −6.80978045157933800495643190449, −6.53597050024361446588302560450, −5.70209754970246892249563765937, −4.85306929357905469613232736326, −3.48520562828069410090501067297, −2.46482257130240376840494731896, −2.10538879884686024812700356306, −1.36909126667306602840940231155, 0,
1.36909126667306602840940231155, 2.10538879884686024812700356306, 2.46482257130240376840494731896, 3.48520562828069410090501067297, 4.85306929357905469613232736326, 5.70209754970246892249563765937, 6.53597050024361446588302560450, 6.80978045157933800495643190449, 7.67823462040903217127011323866