| L(s) = 1 | + 2-s − 1.34·3-s + 4-s + 5-s − 1.34·6-s + 7-s + 8-s − 1.19·9-s + 10-s − 1.34·12-s + 4.31·13-s + 14-s − 1.34·15-s + 16-s − 2.63·17-s − 1.19·18-s + 1.56·19-s + 20-s − 1.34·21-s + 5.45·23-s − 1.34·24-s + 25-s + 4.31·26-s + 5.63·27-s + 28-s + 2.83·29-s − 1.34·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.775·3-s + 0.5·4-s + 0.447·5-s − 0.548·6-s + 0.377·7-s + 0.353·8-s − 0.398·9-s + 0.316·10-s − 0.387·12-s + 1.19·13-s + 0.267·14-s − 0.346·15-s + 0.250·16-s − 0.639·17-s − 0.282·18-s + 0.359·19-s + 0.223·20-s − 0.293·21-s + 1.13·23-s − 0.274·24-s + 0.200·25-s + 0.845·26-s + 1.08·27-s + 0.188·28-s + 0.527·29-s − 0.245·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.000554379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.000554379\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 1.34T + 3T^{2} \) |
| 13 | \( 1 - 4.31T + 13T^{2} \) |
| 17 | \( 1 + 2.63T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 - 5.45T + 23T^{2} \) |
| 29 | \( 1 - 2.83T + 29T^{2} \) |
| 31 | \( 1 - 1.14T + 31T^{2} \) |
| 37 | \( 1 + 0.839T + 37T^{2} \) |
| 41 | \( 1 - 3.47T + 41T^{2} \) |
| 43 | \( 1 + 2.01T + 43T^{2} \) |
| 47 | \( 1 + 8.39T + 47T^{2} \) |
| 53 | \( 1 - 1.40T + 53T^{2} \) |
| 59 | \( 1 - 2.32T + 59T^{2} \) |
| 61 | \( 1 + 5.24T + 61T^{2} \) |
| 67 | \( 1 + 1.60T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 1.14T + 73T^{2} \) |
| 79 | \( 1 - 9.65T + 79T^{2} \) |
| 83 | \( 1 + 2.51T + 83T^{2} \) |
| 89 | \( 1 - 2.19T + 89T^{2} \) |
| 97 | \( 1 + 7.56T + 97T^{2} \) |
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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
−7.61512324741244772217488290096, −6.80322001226660149345022720374, −6.22185267666527319959451103168, −5.77294609610918704339465355540, −4.97715718474848267954870814228, −4.55647032240167897051495014616, −3.46997565712715089435385083093, −2.80708026547922481827297384406, −1.74142416255422947762213437232, −0.821235043728869545470673534759,
0.821235043728869545470673534759, 1.74142416255422947762213437232, 2.80708026547922481827297384406, 3.46997565712715089435385083093, 4.55647032240167897051495014616, 4.97715718474848267954870814228, 5.77294609610918704339465355540, 6.22185267666527319959451103168, 6.80322001226660149345022720374, 7.61512324741244772217488290096
