| L(s) = 1 | + 5.04·2-s + 17.4·4-s + 20.1·5-s + 15.4·7-s + 47.7·8-s + 101.·10-s − 26.9·11-s + 77.8·14-s + 101.·16-s − 23.2·17-s − 45.0·19-s + 351.·20-s − 135.·22-s + 142.·23-s + 279.·25-s + 269.·28-s − 2.29·29-s − 37.7·31-s + 128.·32-s − 117.·34-s + 310.·35-s + 313.·37-s − 227.·38-s + 959.·40-s + 5.86·41-s − 360.·43-s − 470.·44-s + ⋯ |
| L(s) = 1 | + 1.78·2-s + 2.18·4-s + 1.79·5-s + 0.833·7-s + 2.10·8-s + 3.20·10-s − 0.738·11-s + 1.48·14-s + 1.57·16-s − 0.331·17-s − 0.544·19-s + 3.92·20-s − 1.31·22-s + 1.28·23-s + 2.23·25-s + 1.81·28-s − 0.0146·29-s − 0.218·31-s + 0.708·32-s − 0.591·34-s + 1.49·35-s + 1.39·37-s − 0.970·38-s + 3.79·40-s + 0.0223·41-s − 1.27·43-s − 1.61·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(10.63359316\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.63359316\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 5.04T + 8T^{2} \) |
| 5 | \( 1 - 20.1T + 125T^{2} \) |
| 7 | \( 1 - 15.4T + 343T^{2} \) |
| 11 | \( 1 + 26.9T + 1.33e3T^{2} \) |
| 17 | \( 1 + 23.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 45.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 142.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.29T + 2.43e4T^{2} \) |
| 31 | \( 1 + 37.7T + 2.97e4T^{2} \) |
| 37 | \( 1 - 313.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 5.86T + 6.89e4T^{2} \) |
| 43 | \( 1 + 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 209.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 276.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 543.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 205.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 492.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 826.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 66.1T + 3.89e5T^{2} \) |
| 79 | \( 1 - 317.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 141.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 641.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.11e3T + 9.12e5T^{2} \) |
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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
−9.200735286289458063411558782524, −8.163489135623042579262171033336, −6.99204081696878097783024271956, −6.38383360868078794973024257561, −5.48395236519311653313846854671, −5.13602028965703558329245127034, −4.30396540466612687807627626285, −2.91862361283585892027380121044, −2.29495925968687375903221503636, −1.42430141688623934322855723612,
1.42430141688623934322855723612, 2.29495925968687375903221503636, 2.91862361283585892027380121044, 4.30396540466612687807627626285, 5.13602028965703558329245127034, 5.48395236519311653313846854671, 6.38383360868078794973024257561, 6.99204081696878097783024271956, 8.163489135623042579262171033336, 9.200735286289458063411558782524
