| L(s) = 1 | + 2-s − 0.788·3-s + 4-s + 4.07·5-s − 0.788·6-s − 1.81·7-s + 8-s − 2.37·9-s + 4.07·10-s − 6.23·11-s − 0.788·12-s + 1.33·13-s − 1.81·14-s − 3.21·15-s + 16-s − 5.71·17-s − 2.37·18-s − 5.18·19-s + 4.07·20-s + 1.43·21-s − 6.23·22-s + 6.95·23-s − 0.788·24-s + 11.6·25-s + 1.33·26-s + 4.23·27-s − 1.81·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.455·3-s + 0.5·4-s + 1.82·5-s − 0.321·6-s − 0.686·7-s + 0.353·8-s − 0.792·9-s + 1.28·10-s − 1.87·11-s − 0.227·12-s + 0.371·13-s − 0.485·14-s − 0.829·15-s + 0.250·16-s − 1.38·17-s − 0.560·18-s − 1.18·19-s + 0.911·20-s + 0.312·21-s − 1.32·22-s + 1.45·23-s − 0.160·24-s + 2.32·25-s + 0.262·26-s + 0.815·27-s − 0.343·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 2003 | \( 1 - T \) |
| good | 3 | \( 1 + 0.788T + 3T^{2} \) |
| 5 | \( 1 - 4.07T + 5T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 11 | \( 1 + 6.23T + 11T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 + 5.71T + 17T^{2} \) |
| 19 | \( 1 + 5.18T + 19T^{2} \) |
| 23 | \( 1 - 6.95T + 23T^{2} \) |
| 29 | \( 1 + 1.49T + 29T^{2} \) |
| 31 | \( 1 - 4.58T + 31T^{2} \) |
| 37 | \( 1 - 5.10T + 37T^{2} \) |
| 41 | \( 1 + 5.79T + 41T^{2} \) |
| 43 | \( 1 + 6.81T + 43T^{2} \) |
| 47 | \( 1 + 7.99T + 47T^{2} \) |
| 53 | \( 1 + 9.02T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 9.97T + 61T^{2} \) |
| 67 | \( 1 + 11.6T + 67T^{2} \) |
| 71 | \( 1 + 6.38T + 71T^{2} \) |
| 73 | \( 1 + 0.321T + 73T^{2} \) |
| 79 | \( 1 - 6.21T + 79T^{2} \) |
| 83 | \( 1 - 1.21T + 83T^{2} \) |
| 89 | \( 1 + 9.16T + 89T^{2} \) |
| 97 | \( 1 + 9.75T + 97T^{2} \) |
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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
−8.180904725690840267695922771231, −6.86579680112357962373087731544, −6.34761348808563659839736398439, −5.93445068906229479941234948986, −5.05387768982951093874320079535, −4.75116746123688505080519580610, −3.01200509779712599427808221050, −2.69690271895764435862273213133, −1.74970709280918870186242805680, 0,
1.74970709280918870186242805680, 2.69690271895764435862273213133, 3.01200509779712599427808221050, 4.75116746123688505080519580610, 5.05387768982951093874320079535, 5.93445068906229479941234948986, 6.34761348808563659839736398439, 6.86579680112357962373087731544, 8.180904725690840267695922771231
