| L(s) = 1 | − 5.44·2-s + 3.46·3-s + 21.6·4-s + 9.65·5-s − 18.8·6-s − 74.3·8-s − 14.9·9-s − 52.5·10-s + 36.7·11-s + 75.1·12-s + 41.8·13-s + 33.4·15-s + 231.·16-s − 125.·17-s + 81.5·18-s − 43.9·19-s + 209.·20-s − 199.·22-s + 90.6·23-s − 258.·24-s − 31.7·25-s − 228.·26-s − 145.·27-s + 103.·29-s − 182.·30-s − 264.·31-s − 667.·32-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 0.667·3-s + 2.70·4-s + 0.863·5-s − 1.28·6-s − 3.28·8-s − 0.554·9-s − 1.66·10-s + 1.00·11-s + 1.80·12-s + 0.893·13-s + 0.576·15-s + 3.62·16-s − 1.79·17-s + 1.06·18-s − 0.531·19-s + 2.33·20-s − 1.93·22-s + 0.821·23-s − 2.19·24-s − 0.254·25-s − 1.72·26-s − 1.03·27-s + 0.664·29-s − 1.10·30-s − 1.53·31-s − 3.68·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 41 | \( 1 + 41T \) |
| good | 2 | \( 1 + 5.44T + 8T^{2} \) |
| 3 | \( 1 - 3.46T + 27T^{2} \) |
| 5 | \( 1 - 9.65T + 125T^{2} \) |
| 11 | \( 1 - 36.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 125.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 43.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 90.6T + 1.21e4T^{2} \) |
| 29 | \( 1 - 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 264.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 6.96T + 5.06e4T^{2} \) |
| 43 | \( 1 - 537.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 256.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 481.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 468.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 564.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 177.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 9.15T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 390.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 499.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 574.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 745.T + 9.12e5T^{2} \) |
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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
−8.793862328171062471117816738599, −7.955527555866965228057626607842, −6.98567874224132629247960279226, −6.37687941682677742696491490249, −5.72920731086727811850298533030, −3.97162790612643586401549601589, −2.78244269595263498895340476815, −2.07249533208042009028562090985, −1.28721155283444874298983830177, 0,
1.28721155283444874298983830177, 2.07249533208042009028562090985, 2.78244269595263498895340476815, 3.97162790612643586401549601589, 5.72920731086727811850298533030, 6.37687941682677742696491490249, 6.98567874224132629247960279226, 7.955527555866965228057626607842, 8.793862328171062471117816738599
