| L(s) = 1 | + 2.01·2-s + 2.05·4-s + 0.369·7-s + 0.110·8-s + 1.74·11-s + 1.11·13-s + 0.745·14-s − 3.88·16-s − 5.48·17-s − 3.75·19-s + 3.51·22-s − 7.24·23-s + 2.24·26-s + 0.760·28-s − 4.19·29-s + 0.305·31-s − 8.04·32-s − 11.0·34-s − 9.21·37-s − 7.55·38-s + 4.18·41-s + 7.17·43-s + 3.58·44-s − 14.5·46-s − 0.810·47-s − 6.86·49-s + 2.29·52-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 1.02·4-s + 0.139·7-s + 0.0390·8-s + 0.526·11-s + 0.309·13-s + 0.199·14-s − 0.971·16-s − 1.33·17-s − 0.860·19-s + 0.749·22-s − 1.51·23-s + 0.440·26-s + 0.143·28-s − 0.778·29-s + 0.0549·31-s − 1.42·32-s − 1.89·34-s − 1.51·37-s − 1.22·38-s + 0.653·41-s + 1.09·43-s + 0.540·44-s − 2.15·46-s − 0.118·47-s − 0.980·49-s + 0.318·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.01T + 2T^{2} \) |
| 7 | \( 1 - 0.369T + 7T^{2} \) |
| 11 | \( 1 - 1.74T + 11T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 17 | \( 1 + 5.48T + 17T^{2} \) |
| 19 | \( 1 + 3.75T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 - 0.305T + 31T^{2} \) |
| 37 | \( 1 + 9.21T + 37T^{2} \) |
| 41 | \( 1 - 4.18T + 41T^{2} \) |
| 43 | \( 1 - 7.17T + 43T^{2} \) |
| 47 | \( 1 + 0.810T + 47T^{2} \) |
| 53 | \( 1 - 3.91T + 53T^{2} \) |
| 59 | \( 1 - 1.85T + 59T^{2} \) |
| 61 | \( 1 - 9.68T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 3.43T + 73T^{2} \) |
| 79 | \( 1 - 5.69T + 79T^{2} \) |
| 83 | \( 1 + 7.13T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−7.55663638315597215275892301936, −6.75264625461783153051562305932, −6.18876300112314119688786612643, −5.62617170069224722571368750241, −4.68738585153350504679179188020, −4.12251023020267037779448178893, −3.60073772000070482118398263354, −2.47949055142319922225285006990, −1.79610617602725714367260948264, 0,
1.79610617602725714367260948264, 2.47949055142319922225285006990, 3.60073772000070482118398263354, 4.12251023020267037779448178893, 4.68738585153350504679179188020, 5.62617170069224722571368750241, 6.18876300112314119688786612643, 6.75264625461783153051562305932, 7.55663638315597215275892301936
