| L(s) = 1 | + 4.54·2-s + 3·3-s + 12.6·4-s − 6.53·5-s + 13.6·6-s + 21.0·8-s + 9·9-s − 29.6·10-s + 11·11-s + 37.9·12-s − 71.3·13-s − 19.6·15-s − 5.29·16-s − 2.45·17-s + 40.8·18-s − 80.0·19-s − 82.6·20-s + 49.9·22-s + 61.8·23-s + 63.2·24-s − 82.2·25-s − 324.·26-s + 27·27-s − 156.·29-s − 89.0·30-s − 77.7·31-s − 192.·32-s + ⋯ |
| L(s) = 1 | + 1.60·2-s + 0.577·3-s + 1.58·4-s − 0.584·5-s + 0.927·6-s + 0.932·8-s + 0.333·9-s − 0.939·10-s + 0.301·11-s + 0.912·12-s − 1.52·13-s − 0.337·15-s − 0.0827·16-s − 0.0349·17-s + 0.535·18-s − 0.966·19-s − 0.923·20-s + 0.484·22-s + 0.560·23-s + 0.538·24-s − 0.658·25-s − 2.44·26-s + 0.192·27-s − 1.00·29-s − 0.542·30-s − 0.450·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 3T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 - 4.54T + 8T^{2} \) |
| 5 | \( 1 + 6.53T + 125T^{2} \) |
| 13 | \( 1 + 71.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.45T + 4.91e3T^{2} \) |
| 19 | \( 1 + 80.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 61.8T + 1.21e4T^{2} \) |
| 29 | \( 1 + 156.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 77.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 84.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 28.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 352.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 256.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 492.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 3.12T + 2.05e5T^{2} \) |
| 61 | \( 1 - 159.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 521.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 885.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 375.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.37e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 111.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 472.T + 9.12e5T^{2} \) |
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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
−8.560713938987587326142922027532, −7.46275163912024020622073116756, −7.05245640119335833845010909129, −6.01142544859752240193405818277, −5.09753530900518920893912784528, −4.33075835304703981436348690490, −3.70402156966954834669857243192, −2.75007041839759932864232599447, −1.93823171177398946117596892542, 0,
1.93823171177398946117596892542, 2.75007041839759932864232599447, 3.70402156966954834669857243192, 4.33075835304703981436348690490, 5.09753530900518920893912784528, 6.01142544859752240193405818277, 7.05245640119335833845010909129, 7.46275163912024020622073116756, 8.560713938987587326142922027532
