| L(s) = 1 | − 2-s + 4-s + 1.23·5-s + 7-s − 8-s − 1.23·10-s − 11-s − 3.23·13-s − 14-s + 16-s − 2.47·17-s − 7.23·19-s + 1.23·20-s + 22-s − 4·23-s − 3.47·25-s + 3.23·26-s + 28-s − 4.47·29-s + 2·31-s − 32-s + 2.47·34-s + 1.23·35-s + 6.94·37-s + 7.23·38-s − 1.23·40-s + 2.47·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.552·5-s + 0.377·7-s − 0.353·8-s − 0.390·10-s − 0.301·11-s − 0.897·13-s − 0.267·14-s + 0.250·16-s − 0.599·17-s − 1.66·19-s + 0.276·20-s + 0.213·22-s − 0.834·23-s − 0.694·25-s + 0.634·26-s + 0.188·28-s − 0.830·29-s + 0.359·31-s − 0.176·32-s + 0.423·34-s + 0.208·35-s + 1.14·37-s + 1.17·38-s − 0.195·40-s + 0.386·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 + 2.47T + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 2.47T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 + 0.763T + 61T^{2} \) |
| 67 | \( 1 - 11.4T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−9.310701557908322384563188995918, −8.288753828787384299563870958464, −7.80018270076917448928243724360, −6.71955569501021803670288833574, −6.05627586673234339886059505570, −5.01893725709974169098000558031, −4.02874520618029281560150379012, −2.49773139736167420760306064774, −1.84654494520757238864883313035, 0,
1.84654494520757238864883313035, 2.49773139736167420760306064774, 4.02874520618029281560150379012, 5.01893725709974169098000558031, 6.05627586673234339886059505570, 6.71955569501021803670288833574, 7.80018270076917448928243724360, 8.288753828787384299563870958464, 9.310701557908322384563188995918
