| L(s) = 1 | + 1.62·2-s + 0.649·4-s + 1.79·5-s + 0.552·7-s − 2.19·8-s + 2.92·10-s − 4.33·11-s + 2.54·13-s + 0.898·14-s − 4.87·16-s − 2.52·17-s − 5.11·19-s + 1.16·20-s − 7.06·22-s + 3.78·23-s − 1.76·25-s + 4.14·26-s + 0.358·28-s − 0.907·29-s + 0.380·31-s − 3.54·32-s − 4.10·34-s + 0.993·35-s − 5.43·37-s − 8.33·38-s − 3.95·40-s − 5.72·41-s + ⋯ |
| L(s) = 1 | + 1.15·2-s + 0.324·4-s + 0.804·5-s + 0.208·7-s − 0.777·8-s + 0.925·10-s − 1.30·11-s + 0.706·13-s + 0.240·14-s − 1.21·16-s − 0.611·17-s − 1.17·19-s + 0.261·20-s − 1.50·22-s + 0.789·23-s − 0.352·25-s + 0.813·26-s + 0.0678·28-s − 0.168·29-s + 0.0683·31-s − 0.626·32-s − 0.704·34-s + 0.167·35-s − 0.893·37-s − 1.35·38-s − 0.625·40-s − 0.894·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 2 | \( 1 - 1.62T + 2T^{2} \) |
| 5 | \( 1 - 1.79T + 5T^{2} \) |
| 7 | \( 1 - 0.552T + 7T^{2} \) |
| 11 | \( 1 + 4.33T + 11T^{2} \) |
| 13 | \( 1 - 2.54T + 13T^{2} \) |
| 17 | \( 1 + 2.52T + 17T^{2} \) |
| 19 | \( 1 + 5.11T + 19T^{2} \) |
| 23 | \( 1 - 3.78T + 23T^{2} \) |
| 29 | \( 1 + 0.907T + 29T^{2} \) |
| 31 | \( 1 - 0.380T + 31T^{2} \) |
| 37 | \( 1 + 5.43T + 37T^{2} \) |
| 41 | \( 1 + 5.72T + 41T^{2} \) |
| 43 | \( 1 + 9.21T + 43T^{2} \) |
| 47 | \( 1 - 8.81T + 47T^{2} \) |
| 53 | \( 1 - 6.46T + 53T^{2} \) |
| 59 | \( 1 + 3.40T + 59T^{2} \) |
| 61 | \( 1 + 1.06T + 61T^{2} \) |
| 67 | \( 1 + 0.253T + 67T^{2} \) |
| 71 | \( 1 + 11.7T + 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 - 7.76T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 3.09T + 89T^{2} \) |
| 97 | \( 1 + 4.28T + 97T^{2} \) |
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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
−8.019811802198222024987076227157, −6.92442610668198986382344743818, −6.29796589908943429135830866041, −5.59367785691320654328430227134, −5.05285658753605591360240802626, −4.34321419241259436690898003503, −3.42450009235693585688736584636, −2.59036844542573636747102687851, −1.78334772175142983550567202746, 0,
1.78334772175142983550567202746, 2.59036844542573636747102687851, 3.42450009235693585688736584636, 4.34321419241259436690898003503, 5.05285658753605591360240802626, 5.59367785691320654328430227134, 6.29796589908943429135830866041, 6.92442610668198986382344743818, 8.019811802198222024987076227157
