| L(s) = 1 | + 0.206·2-s − 0.197·3-s − 1.95·4-s − 3.82·5-s − 0.0407·6-s − 0.817·8-s − 2.96·9-s − 0.789·10-s + 3.34·11-s + 0.386·12-s + 4.04·13-s + 0.754·15-s + 3.74·16-s + 3.36·17-s − 0.611·18-s + 5.85·19-s + 7.48·20-s + 0.691·22-s − 6.71·23-s + 0.161·24-s + 9.62·25-s + 0.834·26-s + 1.17·27-s − 2.74·29-s + 0.155·30-s − 3.52·31-s + 2.40·32-s + ⋯ |
| L(s) = 1 | + 0.145·2-s − 0.113·3-s − 0.978·4-s − 1.71·5-s − 0.0166·6-s − 0.288·8-s − 0.987·9-s − 0.249·10-s + 1.00·11-s + 0.111·12-s + 1.12·13-s + 0.194·15-s + 0.936·16-s + 0.815·17-s − 0.144·18-s + 1.34·19-s + 1.67·20-s + 0.147·22-s − 1.40·23-s + 0.0329·24-s + 1.92·25-s + 0.163·26-s + 0.226·27-s − 0.510·29-s + 0.0284·30-s − 0.632·31-s + 0.425·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.206T + 2T^{2} \) |
| 3 | \( 1 + 0.197T + 3T^{2} \) |
| 5 | \( 1 + 3.82T + 5T^{2} \) |
| 11 | \( 1 - 3.34T + 11T^{2} \) |
| 13 | \( 1 - 4.04T + 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 19 | \( 1 - 5.85T + 19T^{2} \) |
| 23 | \( 1 + 6.71T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 3.52T + 31T^{2} \) |
| 37 | \( 1 + 3.70T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 + 6.70T + 43T^{2} \) |
| 47 | \( 1 + 1.40T + 47T^{2} \) |
| 53 | \( 1 + 5.85T + 53T^{2} \) |
| 59 | \( 1 + 4.73T + 59T^{2} \) |
| 61 | \( 1 + 0.0753T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 + 2.45T + 71T^{2} \) |
| 73 | \( 1 - 6.88T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 - 1.71T + 97T^{2} \) |
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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
−8.416978661026768842330411438801, −8.050801139562024765516999100207, −7.21664884132490164541104842015, −6.07185278331993914328275672375, −5.40063773839622447074471438149, −4.37929309921839761279337865840, −3.58833626308404023303682920981, −3.34322977504945314639306590132, −1.16932152602294166283564644163, 0,
1.16932152602294166283564644163, 3.34322977504945314639306590132, 3.58833626308404023303682920981, 4.37929309921839761279337865840, 5.40063773839622447074471438149, 6.07185278331993914328275672375, 7.21664884132490164541104842015, 8.050801139562024765516999100207, 8.416978661026768842330411438801
