| L(s) = 1 | − 2.22·2-s − 3.03·4-s + 7.56·5-s − 1.62·7-s + 24.5·8-s − 16.8·10-s + 38.9·11-s − 66.8·13-s + 3.61·14-s − 30.5·16-s − 47.5·17-s + 134.·19-s − 22.9·20-s − 86.8·22-s + 57.2·23-s − 67.7·25-s + 149.·26-s + 4.92·28-s − 44.0·29-s − 56.6·31-s − 128.·32-s + 105.·34-s − 12.2·35-s + 223.·37-s − 300.·38-s + 186.·40-s + 22.2·41-s + ⋯ |
| L(s) = 1 | − 0.787·2-s − 0.379·4-s + 0.676·5-s − 0.0876·7-s + 1.08·8-s − 0.533·10-s + 1.06·11-s − 1.42·13-s + 0.0690·14-s − 0.477·16-s − 0.678·17-s + 1.62·19-s − 0.256·20-s − 0.841·22-s + 0.519·23-s − 0.541·25-s + 1.12·26-s + 0.0332·28-s − 0.281·29-s − 0.328·31-s − 0.710·32-s + 0.534·34-s − 0.0593·35-s + 0.994·37-s − 1.28·38-s + 0.735·40-s + 0.0849·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + 239T \) |
| good | 2 | \( 1 + 2.22T + 8T^{2} \) |
| 5 | \( 1 - 7.56T + 125T^{2} \) |
| 7 | \( 1 + 1.62T + 343T^{2} \) |
| 11 | \( 1 - 38.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 66.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 134.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 57.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 44.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 56.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 223.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 22.2T + 6.89e4T^{2} \) |
| 43 | \( 1 + 232.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 349.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 271.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 304.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 204.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 747.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 136.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 35.8T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.06e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 700.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 289.T + 9.12e5T^{2} \) |
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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
−8.574108498865767014570024176324, −7.50478585227454736564203580306, −7.08431999742808071870455130548, −6.00496483095402317300513354610, −5.08927679923848685601321053107, −4.40690070327018354769346821077, −3.25673731224164320822220015257, −2.03896864217337933248300525058, −1.15502443117811630542522864531, 0,
1.15502443117811630542522864531, 2.03896864217337933248300525058, 3.25673731224164320822220015257, 4.40690070327018354769346821077, 5.08927679923848685601321053107, 6.00496483095402317300513354610, 7.08431999742808071870455130548, 7.50478585227454736564203580306, 8.574108498865767014570024176324
