| L(s) = 1 | − 2·2-s + 1.76·3-s + 4·4-s − 20.7·5-s − 3.52·6-s + 8.76·7-s − 8·8-s − 23.8·9-s + 41.4·10-s − 62.1·11-s + 7.05·12-s − 64.5·13-s − 17.5·14-s − 36.5·15-s + 16·16-s − 46.4·17-s + 47.7·18-s − 82.8·20-s + 15.4·21-s + 124.·22-s − 37.2·23-s − 14.1·24-s + 303.·25-s + 129.·26-s − 89.7·27-s + 35.0·28-s − 66.4·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.339·3-s + 0.5·4-s − 1.85·5-s − 0.239·6-s + 0.473·7-s − 0.353·8-s − 0.884·9-s + 1.30·10-s − 1.70·11-s + 0.169·12-s − 1.37·13-s − 0.334·14-s − 0.628·15-s + 0.250·16-s − 0.662·17-s + 0.625·18-s − 0.926·20-s + 0.160·21-s + 1.20·22-s − 0.338·23-s − 0.119·24-s + 2.43·25-s + 0.973·26-s − 0.639·27-s + 0.236·28-s − 0.425·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1735013670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1735013670\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 1.76T + 27T^{2} \) |
| 5 | \( 1 + 20.7T + 125T^{2} \) |
| 7 | \( 1 - 8.76T + 343T^{2} \) |
| 11 | \( 1 + 62.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 46.4T + 4.91e3T^{2} \) |
| 23 | \( 1 + 37.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 66.4T + 2.43e4T^{2} \) |
| 31 | \( 1 - 112.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 189.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 240.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 168.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 187.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 113.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 185.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 308.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 39.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 350.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 9.60T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 257.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 133.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.19e3T + 9.12e5T^{2} \) |
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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
−10.08731899036230738577507520252, −8.826946109203603545460033086782, −8.174320995878936278126452930216, −7.71001907080689022765428444101, −7.00880172371570034038367843109, −5.37965840513156294929560171619, −4.50455033251725668621558512183, −3.18528066279893655900914981490, −2.35828635850231993145017987838, −0.23990912904581069583215100151,
0.23990912904581069583215100151, 2.35828635850231993145017987838, 3.18528066279893655900914981490, 4.50455033251725668621558512183, 5.37965840513156294929560171619, 7.00880172371570034038367843109, 7.71001907080689022765428444101, 8.174320995878936278126452930216, 8.826946109203603545460033086782, 10.08731899036230738577507520252
