| L(s) = 1 | − 2-s − 3-s + 4-s − 1.11·5-s + 6-s − 1.49·7-s − 8-s + 9-s + 1.11·10-s + 5.62·11-s − 12-s − 0.691·13-s + 1.49·14-s + 1.11·15-s + 16-s + 17-s − 18-s + 2.30·19-s − 1.11·20-s + 1.49·21-s − 5.62·22-s − 2.52·23-s + 24-s − 3.74·25-s + 0.691·26-s − 27-s − 1.49·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.500·5-s + 0.408·6-s − 0.565·7-s − 0.353·8-s + 0.333·9-s + 0.353·10-s + 1.69·11-s − 0.288·12-s − 0.191·13-s + 0.399·14-s + 0.288·15-s + 0.250·16-s + 0.242·17-s − 0.235·18-s + 0.528·19-s − 0.250·20-s + 0.326·21-s − 1.19·22-s − 0.527·23-s + 0.204·24-s − 0.749·25-s + 0.135·26-s − 0.192·27-s − 0.282·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 1.11T + 5T^{2} \) |
| 7 | \( 1 + 1.49T + 7T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 + 0.691T + 13T^{2} \) |
| 19 | \( 1 - 2.30T + 19T^{2} \) |
| 23 | \( 1 + 2.52T + 23T^{2} \) |
| 29 | \( 1 + 6.49T + 29T^{2} \) |
| 31 | \( 1 - 8.03T + 31T^{2} \) |
| 37 | \( 1 + 9.99T + 37T^{2} \) |
| 41 | \( 1 + 5.57T + 41T^{2} \) |
| 43 | \( 1 - 2.88T + 43T^{2} \) |
| 47 | \( 1 + 0.468T + 47T^{2} \) |
| 53 | \( 1 + 0.854T + 53T^{2} \) |
| 61 | \( 1 + 0.0695T + 61T^{2} \) |
| 67 | \( 1 - 4.05T + 67T^{2} \) |
| 71 | \( 1 - 8.28T + 71T^{2} \) |
| 73 | \( 1 + 0.315T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 0.567T + 83T^{2} \) |
| 89 | \( 1 - 3.50T + 89T^{2} \) |
| 97 | \( 1 + 4.50T + 97T^{2} \) |
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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
−7.71492463555877323567606210465, −6.95367878336700452451648393875, −6.48680176528658365243072198802, −5.80862262130066714211826642998, −4.87219854407902581039325742076, −3.82991673342037083950346166001, −3.42746816216111780095342135458, −2.03997354777611283213500793295, −1.11175096201656082744417351548, 0,
1.11175096201656082744417351548, 2.03997354777611283213500793295, 3.42746816216111780095342135458, 3.82991673342037083950346166001, 4.87219854407902581039325742076, 5.80862262130066714211826642998, 6.48680176528658365243072198802, 6.95367878336700452451648393875, 7.71492463555877323567606210465
