| L(s) = 1 | − 2.46·2-s − 5.32·3-s − 1.92·4-s + 15.0·5-s + 13.1·6-s + 24.4·8-s + 1.36·9-s − 37.1·10-s + 53.1·11-s + 10.2·12-s − 52.6·13-s − 80.2·15-s − 44.8·16-s + 118.·17-s − 3.36·18-s + 43.3·19-s − 29.0·20-s − 130.·22-s − 107.·23-s − 130.·24-s + 102.·25-s + 129.·26-s + 136.·27-s + 242.·29-s + 197.·30-s − 171.·31-s − 85.0·32-s + ⋯ |
| L(s) = 1 | − 0.871·2-s − 1.02·3-s − 0.240·4-s + 1.34·5-s + 0.893·6-s + 1.08·8-s + 0.0505·9-s − 1.17·10-s + 1.45·11-s + 0.246·12-s − 1.12·13-s − 1.38·15-s − 0.701·16-s + 1.68·17-s − 0.0440·18-s + 0.523·19-s − 0.324·20-s − 1.26·22-s − 0.977·23-s − 1.10·24-s + 0.818·25-s + 0.978·26-s + 0.973·27-s + 1.55·29-s + 1.20·30-s − 0.995·31-s − 0.470·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.273390184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.273390184\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + 2.46T + 8T^{2} \) |
| 3 | \( 1 + 5.32T + 27T^{2} \) |
| 5 | \( 1 - 15.0T + 125T^{2} \) |
| 11 | \( 1 - 53.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 118.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 43.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 242.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 171.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 33.9T + 5.06e4T^{2} \) |
| 41 | \( 1 - 192.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 306.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 328.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 44.9T + 1.48e5T^{2} \) |
| 59 | \( 1 - 297.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 24.7T + 2.26e5T^{2} \) |
| 67 | \( 1 - 269.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 736.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 89.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 162.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 293.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 274.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 505.T + 9.12e5T^{2} \) |
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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
−8.883269213173209946549662116040, −7.87173342776367861108621817575, −7.07066038215647294894958388447, −6.19482181148066830537385592463, −5.58178160507245126681053262356, −4.93815524181835827305875518371, −3.88452028762262076892579123789, −2.44564747077315025897412080334, −1.32021091119680678937697461117, −0.70331160106890851167570888947,
0.70331160106890851167570888947, 1.32021091119680678937697461117, 2.44564747077315025897412080334, 3.88452028762262076892579123789, 4.93815524181835827305875518371, 5.58178160507245126681053262356, 6.19482181148066830537385592463, 7.07066038215647294894958388447, 7.87173342776367861108621817575, 8.883269213173209946549662116040
