| L(s) = 1 | − 2-s + 1.75·3-s + 4-s − 3.17·5-s − 1.75·6-s − 5.12·7-s − 8-s + 0.0936·9-s + 3.17·10-s + 1.69·11-s + 1.75·12-s + 5.12·14-s − 5.57·15-s + 16-s + 5.66·17-s − 0.0936·18-s + 19-s − 3.17·20-s − 9.01·21-s − 1.69·22-s − 6.00·23-s − 1.75·24-s + 5.05·25-s − 5.11·27-s − 5.12·28-s + 8.22·29-s + 5.57·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.01·3-s + 0.5·4-s − 1.41·5-s − 0.718·6-s − 1.93·7-s − 0.353·8-s + 0.0312·9-s + 1.00·10-s + 0.510·11-s + 0.507·12-s + 1.37·14-s − 1.44·15-s + 0.250·16-s + 1.37·17-s − 0.0220·18-s + 0.229·19-s − 0.709·20-s − 1.96·21-s − 0.360·22-s − 1.25·23-s − 0.359·24-s + 1.01·25-s − 0.983·27-s − 0.968·28-s + 1.52·29-s + 1.01·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.75T + 3T^{2} \) |
| 5 | \( 1 + 3.17T + 5T^{2} \) |
| 7 | \( 1 + 5.12T + 7T^{2} \) |
| 11 | \( 1 - 1.69T + 11T^{2} \) |
| 17 | \( 1 - 5.66T + 17T^{2} \) |
| 23 | \( 1 + 6.00T + 23T^{2} \) |
| 29 | \( 1 - 8.22T + 29T^{2} \) |
| 31 | \( 1 - 6.60T + 31T^{2} \) |
| 37 | \( 1 - 6.76T + 37T^{2} \) |
| 41 | \( 1 - 9.51T + 41T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 + 4.27T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 1.43T + 59T^{2} \) |
| 61 | \( 1 + 9.84T + 61T^{2} \) |
| 67 | \( 1 + 9.48T + 67T^{2} \) |
| 71 | \( 1 - 4.44T + 71T^{2} \) |
| 73 | \( 1 + 7.63T + 73T^{2} \) |
| 79 | \( 1 - 3.47T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 + 15.8T + 89T^{2} \) |
| 97 | \( 1 - 9.57T + 97T^{2} \) |
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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
−7.88475987914478884274896874938, −7.28161847971185780468270667204, −6.32269916265302187369284193552, −5.98469836222254256671325714303, −4.39312202512820865307442170014, −3.72202540293350539750480088453, −3.05669831292509540084954404105, −2.70213053848585426630826241609, −1.03831140411724824594538285183, 0,
1.03831140411724824594538285183, 2.70213053848585426630826241609, 3.05669831292509540084954404105, 3.72202540293350539750480088453, 4.39312202512820865307442170014, 5.98469836222254256671325714303, 6.32269916265302187369284193552, 7.28161847971185780468270667204, 7.88475987914478884274896874938
