| L(s) = 1 | − 2.70·2-s − 2.80·3-s + 5.30·4-s − 0.445·5-s + 7.58·6-s − 7-s − 8.94·8-s + 4.88·9-s + 1.20·10-s − 14.8·12-s − 0.450·13-s + 2.70·14-s + 1.24·15-s + 13.5·16-s − 4.83·17-s − 13.1·18-s + 1.08·19-s − 2.36·20-s + 2.80·21-s + 4.57·23-s + 25.0·24-s − 4.80·25-s + 1.21·26-s − 5.27·27-s − 5.30·28-s + 1.98·29-s − 3.37·30-s + ⋯ |
| L(s) = 1 | − 1.91·2-s − 1.62·3-s + 2.65·4-s − 0.199·5-s + 3.09·6-s − 0.377·7-s − 3.16·8-s + 1.62·9-s + 0.380·10-s − 4.30·12-s − 0.125·13-s + 0.722·14-s + 0.322·15-s + 3.38·16-s − 1.17·17-s − 3.10·18-s + 0.249·19-s − 0.528·20-s + 0.612·21-s + 0.953·23-s + 5.12·24-s − 0.960·25-s + 0.239·26-s − 1.01·27-s − 1.00·28-s + 0.368·29-s − 0.616·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.70T + 2T^{2} \) |
| 3 | \( 1 + 2.80T + 3T^{2} \) |
| 5 | \( 1 + 0.445T + 5T^{2} \) |
| 13 | \( 1 + 0.450T + 13T^{2} \) |
| 17 | \( 1 + 4.83T + 17T^{2} \) |
| 19 | \( 1 - 1.08T + 19T^{2} \) |
| 23 | \( 1 - 4.57T + 23T^{2} \) |
| 29 | \( 1 - 1.98T + 29T^{2} \) |
| 31 | \( 1 - 8.25T + 31T^{2} \) |
| 37 | \( 1 - 7.31T + 37T^{2} \) |
| 41 | \( 1 + 1.77T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 - 1.02T + 47T^{2} \) |
| 53 | \( 1 + 3.57T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 4.92T + 61T^{2} \) |
| 67 | \( 1 + 6.18T + 67T^{2} \) |
| 71 | \( 1 + 5.92T + 71T^{2} \) |
| 73 | \( 1 - 1.65T + 73T^{2} \) |
| 79 | \( 1 - 3.60T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 + 5.30T + 97T^{2} \) |
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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
−9.751779961756953981039256928860, −9.182990718062976109557982456775, −8.062027120397730016061340805227, −7.21892546947961434232551994462, −6.46106355462414508342917637421, −5.91162918799560542352468053083, −4.52848967259081485390904781680, −2.68154648708726456744149963679, −1.14335703309485490666222457065, 0,
1.14335703309485490666222457065, 2.68154648708726456744149963679, 4.52848967259081485390904781680, 5.91162918799560542352468053083, 6.46106355462414508342917637421, 7.21892546947961434232551994462, 8.062027120397730016061340805227, 9.182990718062976109557982456775, 9.751779961756953981039256928860
