| L(s) = 1 | + 7-s − 6·13-s + 2·17-s − 8·19-s − 8·23-s − 2·29-s − 4·31-s − 2·37-s + 6·41-s − 4·43-s − 8·47-s + 49-s − 10·53-s − 4·59-s + 2·61-s − 4·67-s − 12·71-s + 2·73-s − 8·79-s − 4·83-s + 6·89-s − 6·91-s + 18·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.66·13-s + 0.485·17-s − 1.83·19-s − 1.66·23-s − 0.371·29-s − 0.718·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s − 1.37·53-s − 0.520·59-s + 0.256·61-s − 0.488·67-s − 1.42·71-s + 0.234·73-s − 0.900·79-s − 0.439·83-s + 0.635·89-s − 0.628·91-s + 1.82·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.27505714220885, −13.95014331484905, −13.02944221238663, −12.79836665788297, −12.33704832797216, −11.77791502178915, −11.42194294285445, −10.70786336567477, −10.26569510990014, −9.904504990954419, −9.316006753834759, −8.784947149868005, −8.165414521534540, −7.742398732039861, −7.340874784711406, −6.653543381571428, −6.089528915490382, −5.660232828376618, −4.804697256890094, −4.613340882476589, −3.915980546597714, −3.282190342285048, −2.440932924700599, −2.040898165843960, −1.433245414021026, 0, 0,
1.433245414021026, 2.040898165843960, 2.440932924700599, 3.282190342285048, 3.915980546597714, 4.613340882476589, 4.804697256890094, 5.660232828376618, 6.089528915490382, 6.653543381571428, 7.340874784711406, 7.742398732039861, 8.165414521534540, 8.784947149868005, 9.316006753834759, 9.904504990954419, 10.26569510990014, 10.70786336567477, 11.42194294285445, 11.77791502178915, 12.33704832797216, 12.79836665788297, 13.02944221238663, 13.95014331484905, 14.27505714220885
