| L(s) = 1 | + 0.297·2-s + 3-s − 1.91·4-s + 1.09·5-s + 0.297·6-s − 7-s − 1.16·8-s + 9-s + 0.324·10-s + 2.37·11-s − 1.91·12-s − 2.19·13-s − 0.297·14-s + 1.09·15-s + 3.47·16-s + 1.34·17-s + 0.297·18-s + 0.625·19-s − 2.08·20-s − 21-s + 0.705·22-s − 9.41·23-s − 1.16·24-s − 3.80·25-s − 0.651·26-s + 27-s + 1.91·28-s + ⋯ |
| L(s) = 1 | + 0.210·2-s + 0.577·3-s − 0.955·4-s + 0.488·5-s + 0.121·6-s − 0.377·7-s − 0.410·8-s + 0.333·9-s + 0.102·10-s + 0.716·11-s − 0.551·12-s − 0.608·13-s − 0.0793·14-s + 0.282·15-s + 0.869·16-s + 0.326·17-s + 0.0700·18-s + 0.143·19-s − 0.467·20-s − 0.218·21-s + 0.150·22-s − 1.96·23-s − 0.237·24-s − 0.761·25-s − 0.127·26-s + 0.192·27-s + 0.361·28-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 | \( 1 - 0.297T + 2T^{2} \) |
| 5 | \( 1 - 1.09T + 5T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 + 2.19T + 13T^{2} \) |
| 17 | \( 1 - 1.34T + 17T^{2} \) |
| 19 | \( 1 - 0.625T + 19T^{2} \) |
| 23 | \( 1 + 9.41T + 23T^{2} \) |
| 29 | \( 1 - 5.47T + 29T^{2} \) |
| 31 | \( 1 - 6.16T + 31T^{2} \) |
| 37 | \( 1 + 8.36T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 9.78T + 43T^{2} \) |
| 47 | \( 1 + 9.06T + 47T^{2} \) |
| 53 | \( 1 + 8.01T + 53T^{2} \) |
| 59 | \( 1 - 10.1T + 59T^{2} \) |
| 61 | \( 1 + 3.04T + 61T^{2} \) |
| 67 | \( 1 - 2.31T + 67T^{2} \) |
| 71 | \( 1 + 3.85T + 71T^{2} \) |
| 73 | \( 1 - 7.01T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 - 13.4T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 - 7.80T + 97T^{2} \) |
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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
−7.70733854290238382604459822309, −6.61955721446633224373762281213, −6.14915613910169104100526514465, −5.31758613410590602179216723941, −4.56750430607183757352300404970, −3.89439845105804595308043726891, −3.24463358020075146251842497080, −2.30084298840429284749411741404, −1.32765094385736132141639423707, 0,
1.32765094385736132141639423707, 2.30084298840429284749411741404, 3.24463358020075146251842497080, 3.89439845105804595308043726891, 4.56750430607183757352300404970, 5.31758613410590602179216723941, 6.14915613910169104100526514465, 6.61955721446633224373762281213, 7.70733854290238382604459822309
