L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 4·11-s − 6·13-s + 16-s + 2·17-s − 4·19-s + 20-s − 4·22-s + 8·23-s + 25-s + 6·26-s − 6·29-s + 31-s − 32-s − 2·34-s − 2·37-s + 4·38-s − 40-s + 10·41-s − 4·43-s + 4·44-s − 8·46-s − 50-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.20·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.223·20-s − 0.852·22-s + 1.66·23-s + 1/5·25-s + 1.17·26-s − 1.11·29-s + 0.179·31-s − 0.176·32-s − 0.342·34-s − 0.328·37-s + 0.648·38-s − 0.158·40-s + 1.56·41-s − 0.609·43-s + 0.603·44-s − 1.17·46-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136710 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−13.68857106868605, −13.10818724587253, −12.58096893133845, −12.22598143890069, −11.78756756616428, −11.09163315739443, −10.84267768104473, −10.20124840078902, −9.672072172963575, −9.343927485061639, −8.977980584854504, −8.441195284723397, −7.711863831874515, −7.321503742648290, −6.780151943441869, −6.510277121306070, −5.575022248840204, −5.428247166800758, −4.485928433887297, −4.180397865538745, −3.239239904931330, −2.768692476539447, −2.115377471751269, −1.530186622159863, −0.8564640589477164, 0,
0.8564640589477164, 1.530186622159863, 2.115377471751269, 2.768692476539447, 3.239239904931330, 4.180397865538745, 4.485928433887297, 5.428247166800758, 5.575022248840204, 6.510277121306070, 6.780151943441869, 7.321503742648290, 7.711863831874515, 8.441195284723397, 8.977980584854504, 9.343927485061639, 9.672072172963575, 10.20124840078902, 10.84267768104473, 11.09163315739443, 11.78756756616428, 12.22598143890069, 12.58096893133845, 13.10818724587253, 13.68857106868605