| L(s) = 1 | − 2-s − 0.771·3-s + 4-s + 0.437·5-s + 0.771·6-s − 4.61·7-s − 8-s − 2.40·9-s − 0.437·10-s − 1.80·11-s − 0.771·12-s − 6.12·13-s + 4.61·14-s − 0.336·15-s + 16-s + 2.20·17-s + 2.40·18-s − 1.36·19-s + 0.437·20-s + 3.55·21-s + 1.80·22-s + 0.264·23-s + 0.771·24-s − 4.80·25-s + 6.12·26-s + 4.16·27-s − 4.61·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.445·3-s + 0.5·4-s + 0.195·5-s + 0.314·6-s − 1.74·7-s − 0.353·8-s − 0.801·9-s − 0.138·10-s − 0.543·11-s − 0.222·12-s − 1.69·13-s + 1.23·14-s − 0.0870·15-s + 0.250·16-s + 0.534·17-s + 0.566·18-s − 0.312·19-s + 0.0977·20-s + 0.775·21-s + 0.384·22-s + 0.0552·23-s + 0.157·24-s − 0.961·25-s + 1.20·26-s + 0.802·27-s − 0.871·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1783927535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1783927535\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + 0.771T + 3T^{2} \) |
| 5 | \( 1 - 0.437T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 + 1.80T + 11T^{2} \) |
| 13 | \( 1 + 6.12T + 13T^{2} \) |
| 17 | \( 1 - 2.20T + 17T^{2} \) |
| 19 | \( 1 + 1.36T + 19T^{2} \) |
| 23 | \( 1 - 0.264T + 23T^{2} \) |
| 29 | \( 1 + 5.46T + 29T^{2} \) |
| 31 | \( 1 + 6.80T + 31T^{2} \) |
| 41 | \( 1 + 9.16T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 - 6.57T + 47T^{2} \) |
| 53 | \( 1 - 8.81T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 3.60T + 61T^{2} \) |
| 67 | \( 1 - 2.02T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 7.17T + 73T^{2} \) |
| 79 | \( 1 + 12.0T + 79T^{2} \) |
| 83 | \( 1 + 8.24T + 83T^{2} \) |
| 89 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + 4.11T + 97T^{2} \) |
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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
−8.983965051680315066163448298597, −8.097584914603320397726306185330, −7.15067429084720348214824483907, −6.76381049221564551613615625102, −5.60392241099342931126296534742, −5.44864426149933870638332718647, −3.83815775266557924767245561414, −2.92237976467930596572244071911, −2.18301704180241458142084177434, −0.27197994474007960697630656115,
0.27197994474007960697630656115, 2.18301704180241458142084177434, 2.92237976467930596572244071911, 3.83815775266557924767245561414, 5.44864426149933870638332718647, 5.60392241099342931126296534742, 6.76381049221564551613615625102, 7.15067429084720348214824483907, 8.097584914603320397726306185330, 8.983965051680315066163448298597
