| L(s) = 1 | + 1.15·2-s − 6.67·4-s + 21.2·5-s + 31.2·7-s − 16.8·8-s + 24.3·10-s + 17.3·11-s + 35.9·14-s + 33.9·16-s − 89.1·17-s + 80.6·19-s − 141.·20-s + 19.9·22-s + 149.·23-s + 324.·25-s − 208.·28-s − 6.30·29-s − 78.3·31-s + 174.·32-s − 102.·34-s + 662.·35-s − 39.2·37-s + 92.7·38-s − 358.·40-s + 330.·41-s − 198.·43-s − 116.·44-s + ⋯ |
| L(s) = 1 | + 0.406·2-s − 0.834·4-s + 1.89·5-s + 1.68·7-s − 0.746·8-s + 0.771·10-s + 0.476·11-s + 0.686·14-s + 0.530·16-s − 1.27·17-s + 0.973·19-s − 1.58·20-s + 0.193·22-s + 1.35·23-s + 2.59·25-s − 1.40·28-s − 0.0403·29-s − 0.453·31-s + 0.962·32-s − 0.517·34-s + 3.20·35-s − 0.174·37-s + 0.396·38-s − 1.41·40-s + 1.25·41-s − 0.705·43-s − 0.397·44-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
| \(L(2)\) |
\(\approx\) |
\(4.288787010\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.288787010\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 1.15T + 8T^{2} \) |
| 5 | \( 1 - 21.2T + 125T^{2} \) |
| 7 | \( 1 - 31.2T + 343T^{2} \) |
| 11 | \( 1 - 17.3T + 1.33e3T^{2} \) |
| 17 | \( 1 + 89.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 80.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 149.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 6.30T + 2.43e4T^{2} \) |
| 31 | \( 1 + 78.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 39.2T + 5.06e4T^{2} \) |
| 41 | \( 1 - 330.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 198.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 246.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 600.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 709.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 472.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 331.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 472.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 651.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 240.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 538.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 673.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 468.T + 9.12e5T^{2} \) |
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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
−9.016980420972683753929066208346, −8.689613990624009705390497193448, −7.46112726171671319558696310326, −6.43266019645422729658834170894, −5.54819594394532261663123736119, −5.02336735188402604402311066626, −4.38007235485752507054225043221, −2.92199567997045115831728419778, −1.83837878524894448354377368804, −1.06222283810069283272261601038,
1.06222283810069283272261601038, 1.83837878524894448354377368804, 2.92199567997045115831728419778, 4.38007235485752507054225043221, 5.02336735188402604402311066626, 5.54819594394532261663123736119, 6.43266019645422729658834170894, 7.46112726171671319558696310326, 8.689613990624009705390497193448, 9.016980420972683753929066208346
