| L(s) = 1 | − 4.24·2-s + 9.99·4-s − 19.9·5-s − 20.9·7-s − 8.48·8-s + 84.7·10-s − 16.0·11-s − 34.9·13-s + 88.9·14-s − 44.0·16-s + 17·17-s − 80.8·19-s − 199.·20-s + 68.0·22-s + 115.·23-s + 273.·25-s + 148.·26-s − 209.·28-s − 154.·29-s + 299.·31-s + 254.·32-s − 72.1·34-s + 418.·35-s + 315.·37-s + 343.·38-s + 169.·40-s − 132.·41-s + ⋯ |
| L(s) = 1 | − 1.49·2-s + 1.24·4-s − 1.78·5-s − 1.13·7-s − 0.374·8-s + 2.67·10-s − 0.439·11-s − 0.745·13-s + 1.69·14-s − 0.687·16-s + 0.242·17-s − 0.976·19-s − 2.23·20-s + 0.659·22-s + 1.05·23-s + 2.19·25-s + 1.11·26-s − 1.41·28-s − 0.986·29-s + 1.73·31-s + 1.40·32-s − 0.363·34-s + 2.02·35-s + 1.40·37-s + 1.46·38-s + 0.669·40-s − 0.503·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2406529109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2406529109\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 - 17T \) |
| good | 2 | \( 1 + 4.24T + 8T^{2} \) |
| 5 | \( 1 + 19.9T + 125T^{2} \) |
| 7 | \( 1 + 20.9T + 343T^{2} \) |
| 11 | \( 1 + 16.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.9T + 2.19e3T^{2} \) |
| 19 | \( 1 + 80.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 115.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 299.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 315.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 132.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 23.1T + 7.95e4T^{2} \) |
| 47 | \( 1 + 260.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 629.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 461.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 789.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 686.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 484.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 254T + 4.93e5T^{2} \) |
| 83 | \( 1 + 548.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 925.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 732.T + 9.12e5T^{2} \) |
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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
−12.22170116790455449887191151972, −11.26081770561853188730511335571, −10.35739599009238982545951259323, −9.352109203467599142255455354035, −8.311378759011161935140592806264, −7.54789780082948615244293211153, −6.66345805682594338967064323612, −4.46644841541842097360227862579, −2.94281767607907934638656114645, −0.47162354457082440620620040855,
0.47162354457082440620620040855, 2.94281767607907934638656114645, 4.46644841541842097360227862579, 6.66345805682594338967064323612, 7.54789780082948615244293211153, 8.311378759011161935140592806264, 9.352109203467599142255455354035, 10.35739599009238982545951259323, 11.26081770561853188730511335571, 12.22170116790455449887191151972
