| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s + 29.3·7-s + 8·8-s + 10·10-s + 38.1·11-s − 22.5·13-s + 58.7·14-s + 16·16-s + 104.·17-s + 141.·19-s + 20·20-s + 76.3·22-s + 23·23-s + 25·25-s − 45.0·26-s + 117.·28-s − 241.·29-s + 99.2·31-s + 32·32-s + 208.·34-s + 146.·35-s + 59.9·37-s + 283.·38-s + 40·40-s − 249.·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.447·5-s + 1.58·7-s + 0.353·8-s + 0.316·10-s + 1.04·11-s − 0.480·13-s + 1.12·14-s + 0.250·16-s + 1.48·17-s + 1.71·19-s + 0.223·20-s + 0.739·22-s + 0.208·23-s + 0.200·25-s − 0.340·26-s + 0.792·28-s − 1.54·29-s + 0.575·31-s + 0.176·32-s + 1.04·34-s + 0.709·35-s + 0.266·37-s + 1.21·38-s + 0.158·40-s − 0.949·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.023261765\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.023261765\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 - 29.3T + 343T^{2} \) |
| 11 | \( 1 - 38.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 22.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 141.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 241.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 99.2T + 2.97e4T^{2} \) |
| 37 | \( 1 - 59.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 249.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 163.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 205.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 491.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 433.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 660.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 323.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 893.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 196.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 500.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 800.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 729.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.13e3T + 9.12e5T^{2} \) |
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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
−8.743828757352141440499670550334, −7.66586539706973244309077012790, −7.39844765930366833644169505522, −6.20321641463877771912974774381, −5.34644296640410737828141519068, −4.94860856036836182974594273501, −3.88793907889407981278967751489, −2.99388554792195916350940846422, −1.70758839601372981000316093954, −1.17443532375049882395611904096,
1.17443532375049882395611904096, 1.70758839601372981000316093954, 2.99388554792195916350940846422, 3.88793907889407981278967751489, 4.94860856036836182974594273501, 5.34644296640410737828141519068, 6.20321641463877771912974774381, 7.39844765930366833644169505522, 7.66586539706973244309077012790, 8.743828757352141440499670550334
