| L(s) = 1 | + 1.42·3-s − 11.4·5-s + 7·7-s − 24.9·9-s + 39.8·11-s + 76.8·13-s − 16.2·15-s − 17·17-s − 80.6·19-s + 9.94·21-s − 135.·23-s + 6.23·25-s − 73.8·27-s + 229.·29-s − 76.2·31-s + 56.6·33-s − 80.1·35-s + 294.·37-s + 109.·39-s − 283.·41-s + 267.·43-s + 286.·45-s + 538.·47-s + 49·49-s − 24.1·51-s − 506.·53-s − 456.·55-s + ⋯ |
| L(s) = 1 | + 0.273·3-s − 1.02·5-s + 0.377·7-s − 0.925·9-s + 1.09·11-s + 1.63·13-s − 0.280·15-s − 0.242·17-s − 0.974·19-s + 0.103·21-s − 1.23·23-s + 0.0499·25-s − 0.526·27-s + 1.47·29-s − 0.441·31-s + 0.298·33-s − 0.387·35-s + 1.30·37-s + 0.448·39-s − 1.08·41-s + 0.947·43-s + 0.948·45-s + 1.67·47-s + 0.142·49-s − 0.0662·51-s − 1.31·53-s − 1.11·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.846838513\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.846838513\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 17 | \( 1 + 17T \) |
| good | 3 | \( 1 - 1.42T + 27T^{2} \) |
| 5 | \( 1 + 11.4T + 125T^{2} \) |
| 11 | \( 1 - 39.8T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.8T + 2.19e3T^{2} \) |
| 19 | \( 1 + 80.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 135.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 76.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 294.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 283.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 267.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 538.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 506.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 304.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 163.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 457.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 1.01e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 757.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 880.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.41e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 788.T + 9.12e5T^{2} \) |
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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
−8.520242617454250842895431511996, −8.374411641921586433955277524686, −7.45184303270091667358786623633, −6.30312738035565900836246859495, −5.93508393842045276403444479844, −4.40808674474259950010003024881, −3.97721109014123907406714136745, −3.07877023969157672348997434528, −1.81688077366674460346483002012, −0.62882948794511988042540609165,
0.62882948794511988042540609165, 1.81688077366674460346483002012, 3.07877023969157672348997434528, 3.97721109014123907406714136745, 4.40808674474259950010003024881, 5.93508393842045276403444479844, 6.30312738035565900836246859495, 7.45184303270091667358786623633, 8.374411641921586433955277524686, 8.520242617454250842895431511996
