| L(s) = 1 | − 5·7-s + 11-s + 4·13-s + 6·17-s − 3·19-s − 7·23-s + 8·29-s + 31-s − 8·37-s + 43-s + 18·49-s − 53-s + 8·59-s + 2·61-s + 10·67-s − 9·71-s − 11·73-s − 5·77-s − 3·79-s − 2·83-s − 9·89-s − 20·91-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.88·7-s + 0.301·11-s + 1.10·13-s + 1.45·17-s − 0.688·19-s − 1.45·23-s + 1.48·29-s + 0.179·31-s − 1.31·37-s + 0.152·43-s + 18/7·49-s − 0.137·53-s + 1.04·59-s + 0.256·61-s + 1.22·67-s − 1.06·71-s − 1.28·73-s − 0.569·77-s − 0.337·79-s − 0.219·83-s − 0.953·89-s − 2.09·91-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.95492496118764, −13.32356581510854, −12.95331405892754, −12.29673417453612, −12.18330510830488, −11.59234399626145, −10.83798100972066, −10.24068949233098, −10.03328225640309, −9.665168218185440, −8.868123320653603, −8.538393514024363, −8.063268977523271, −7.241098112491154, −6.821766864804617, −6.285906976813884, −5.913541901636040, −5.519014584427692, −4.562049531260604, −3.892661919114929, −3.565902385172908, −3.036076585696034, −2.410609649921518, −1.507525480782015, −0.8013357528290574, 0,
0.8013357528290574, 1.507525480782015, 2.410609649921518, 3.036076585696034, 3.565902385172908, 3.892661919114929, 4.562049531260604, 5.519014584427692, 5.913541901636040, 6.285906976813884, 6.821766864804617, 7.241098112491154, 8.063268977523271, 8.538393514024363, 8.868123320653603, 9.665168218185440, 10.03328225640309, 10.24068949233098, 10.83798100972066, 11.59234399626145, 12.18330510830488, 12.29673417453612, 12.95331405892754, 13.32356581510854, 13.95492496118764
