| L(s) = 1 | − 2.78·2-s − 0.518·3-s + 5.77·4-s − 0.346·5-s + 1.44·6-s + 3.25·7-s − 10.5·8-s − 2.73·9-s + 0.966·10-s + 11-s − 2.99·12-s − 1.09·13-s − 9.06·14-s + 0.179·15-s + 17.7·16-s + 2.36·17-s + 7.61·18-s + 2.68·19-s − 2.00·20-s − 1.68·21-s − 2.78·22-s − 4.44·23-s + 5.45·24-s − 4.87·25-s + 3.05·26-s + 2.97·27-s + 18.7·28-s + ⋯ |
| L(s) = 1 | − 1.97·2-s − 0.299·3-s + 2.88·4-s − 0.154·5-s + 0.590·6-s + 1.22·7-s − 3.72·8-s − 0.910·9-s + 0.305·10-s + 0.301·11-s − 0.864·12-s − 0.304·13-s − 2.42·14-s + 0.0464·15-s + 4.44·16-s + 0.572·17-s + 1.79·18-s + 0.615·19-s − 0.447·20-s − 0.368·21-s − 0.594·22-s − 0.926·23-s + 1.11·24-s − 0.975·25-s + 0.599·26-s + 0.571·27-s + 3.54·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 + 2.78T + 2T^{2} \) |
| 3 | \( 1 + 0.518T + 3T^{2} \) |
| 5 | \( 1 + 0.346T + 5T^{2} \) |
| 7 | \( 1 - 3.25T + 7T^{2} \) |
| 13 | \( 1 + 1.09T + 13T^{2} \) |
| 17 | \( 1 - 2.36T + 17T^{2} \) |
| 19 | \( 1 - 2.68T + 19T^{2} \) |
| 23 | \( 1 + 4.44T + 23T^{2} \) |
| 29 | \( 1 + 10.0T + 29T^{2} \) |
| 31 | \( 1 + 5.30T + 31T^{2} \) |
| 37 | \( 1 - 7.22T + 37T^{2} \) |
| 41 | \( 1 + 5.64T + 41T^{2} \) |
| 43 | \( 1 + 6.57T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 2.35T + 53T^{2} \) |
| 59 | \( 1 + 5.75T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 - 3.26T + 71T^{2} \) |
| 73 | \( 1 + 6.82T + 73T^{2} \) |
| 79 | \( 1 - 2.76T + 79T^{2} \) |
| 83 | \( 1 - 4.68T + 83T^{2} \) |
| 89 | \( 1 - 9.75T + 89T^{2} \) |
| 97 | \( 1 + 9.08T + 97T^{2} \) |
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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
−9.210027536443408095973083040735, −8.198264260570586277254975565129, −7.85686511708107859530951284243, −7.13688746067421851995748852980, −5.96772710376690125097508350114, −5.40414537445499838131315780649, −3.63745430224985741312244280823, −2.32825190754642338436456808657, −1.43427730395125939752103080095, 0,
1.43427730395125939752103080095, 2.32825190754642338436456808657, 3.63745430224985741312244280823, 5.40414537445499838131315780649, 5.96772710376690125097508350114, 7.13688746067421851995748852980, 7.85686511708107859530951284243, 8.198264260570586277254975565129, 9.210027536443408095973083040735
