L(s) = 1 | − 2-s + 0.876·3-s + 4-s + 2.33·5-s − 0.876·6-s − 0.00591·7-s − 8-s − 2.23·9-s − 2.33·10-s − 1.45·11-s + 0.876·12-s + 0.647·13-s + 0.00591·14-s + 2.04·15-s + 16-s − 1.62·17-s + 2.23·18-s + 19-s + 2.33·20-s − 0.00518·21-s + 1.45·22-s − 1.78·23-s − 0.876·24-s + 0.469·25-s − 0.647·26-s − 4.58·27-s − 0.00591·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.505·3-s + 0.5·4-s + 1.04·5-s − 0.357·6-s − 0.00223·7-s − 0.353·8-s − 0.744·9-s − 0.739·10-s − 0.439·11-s + 0.252·12-s + 0.179·13-s + 0.00158·14-s + 0.529·15-s + 0.250·16-s − 0.395·17-s + 0.526·18-s + 0.229·19-s + 0.522·20-s − 0.00113·21-s + 0.310·22-s − 0.372·23-s − 0.178·24-s + 0.0939·25-s − 0.126·26-s − 0.882·27-s − 0.00111·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
good | 3 | \( 1 - 0.876T + 3T^{2} \) |
| 5 | \( 1 - 2.33T + 5T^{2} \) |
| 7 | \( 1 + 0.00591T + 7T^{2} \) |
| 11 | \( 1 + 1.45T + 11T^{2} \) |
| 13 | \( 1 - 0.647T + 13T^{2} \) |
| 17 | \( 1 + 1.62T + 17T^{2} \) |
| 23 | \( 1 + 1.78T + 23T^{2} \) |
| 29 | \( 1 - 1.62T + 29T^{2} \) |
| 31 | \( 1 + 2.43T + 31T^{2} \) |
| 37 | \( 1 - 5.87T + 37T^{2} \) |
| 41 | \( 1 - 1.25T + 41T^{2} \) |
| 43 | \( 1 - 0.185T + 43T^{2} \) |
| 47 | \( 1 - 1.39T + 47T^{2} \) |
| 53 | \( 1 - 0.866T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 + 3.52T + 71T^{2} \) |
| 73 | \( 1 - 15.8T + 73T^{2} \) |
| 79 | \( 1 + 8.88T + 79T^{2} \) |
| 83 | \( 1 - 7.78T + 83T^{2} \) |
| 89 | \( 1 - 7.78T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.83637797846452790422030943389, −6.79571525813870143560497019571, −6.11883286498157581513105624047, −5.63665751402542500167634243036, −4.78352700591051676771932070534, −3.66590860746541360917363717870, −2.78156920727764782244589663190, −2.24828372012648938307674064502, −1.37792384652588032581972951799, 0,
1.37792384652588032581972951799, 2.24828372012648938307674064502, 2.78156920727764782244589663190, 3.66590860746541360917363717870, 4.78352700591051676771932070534, 5.63665751402542500167634243036, 6.11883286498157581513105624047, 6.79571525813870143560497019571, 7.83637797846452790422030943389