| L(s) = 1 | − 2·7-s − 11-s − 6·13-s + 6·17-s − 2·19-s − 8·23-s − 5·25-s − 2·29-s − 4·31-s + 2·37-s + 10·41-s − 6·43-s + 4·47-s − 3·49-s − 4·53-s − 4·59-s − 2·61-s − 8·67-s − 12·71-s − 2·73-s + 2·77-s + 14·79-s + 4·83-s + 12·91-s + 2·97-s + 14·101-s − 16·103-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 0.301·11-s − 1.66·13-s + 1.45·17-s − 0.458·19-s − 1.66·23-s − 25-s − 0.371·29-s − 0.718·31-s + 0.328·37-s + 1.56·41-s − 0.914·43-s + 0.583·47-s − 3/7·49-s − 0.549·53-s − 0.520·59-s − 0.256·61-s − 0.977·67-s − 1.42·71-s − 0.234·73-s + 0.227·77-s + 1.57·79-s + 0.439·83-s + 1.25·91-s + 0.203·97-s + 1.39·101-s − 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.869950611494452353243047717437, −9.265412541977345085929379052417, −7.85434660890847189459509785541, −7.50735959345867726619243932720, −6.23628274432277747584412267863, −5.50741629586206527704817119804, −4.33765166705823820039761227894, −3.23609115598580461776406214489, −2.08943186270177677187465865509, 0,
2.08943186270177677187465865509, 3.23609115598580461776406214489, 4.33765166705823820039761227894, 5.50741629586206527704817119804, 6.23628274432277747584412267863, 7.50735959345867726619243932720, 7.85434660890847189459509785541, 9.265412541977345085929379052417, 9.869950611494452353243047717437
