| L(s) = 1 | − 2-s + 3·3-s + 4-s − 5-s − 3·6-s + 2·7-s − 8-s + 6·9-s + 10-s − 4·11-s + 3·12-s − 3·13-s − 2·14-s − 3·15-s + 16-s + 17-s − 6·18-s + 3·19-s − 20-s + 6·21-s + 4·22-s − 6·23-s − 3·24-s + 25-s + 3·26-s + 9·27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s + 0.755·7-s − 0.353·8-s + 2·9-s + 0.316·10-s − 1.20·11-s + 0.866·12-s − 0.832·13-s − 0.534·14-s − 0.774·15-s + 1/4·16-s + 0.242·17-s − 1.41·18-s + 0.688·19-s − 0.223·20-s + 1.30·21-s + 0.852·22-s − 1.25·23-s − 0.612·24-s + 1/5·25-s + 0.588·26-s + 1.73·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.336025674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.336025674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 3 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.83373639226606959528283639348, −11.77392021354054025159681104411, −10.36366338935225999926837372006, −9.673175371223344610087496327723, −8.338175486309026415938497793980, −8.047148785250349636560357968225, −7.12227398289357983628483830063, −4.92015064777793909582073169694, −3.29460314858197068359451412690, −2.08881038737404835754425957871,
2.08881038737404835754425957871, 3.29460314858197068359451412690, 4.92015064777793909582073169694, 7.12227398289357983628483830063, 8.047148785250349636560357968225, 8.338175486309026415938497793980, 9.673175371223344610087496327723, 10.36366338935225999926837372006, 11.77392021354054025159681104411, 12.83373639226606959528283639348
