| L(s) = 1 | − 2-s + 4-s − 2·5-s − 7-s − 8-s + 2·10-s + 4·11-s − 2·13-s + 14-s + 16-s + 4·19-s − 2·20-s − 4·22-s + 8·23-s − 25-s + 2·26-s − 28-s + 6·29-s − 32-s + 2·35-s + 2·37-s − 4·38-s + 2·40-s + 10·41-s − 4·43-s + 4·44-s − 8·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.894·5-s − 0.377·7-s − 0.353·8-s + 0.632·10-s + 1.20·11-s − 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.917·19-s − 0.447·20-s − 0.852·22-s + 1.66·23-s − 1/5·25-s + 0.392·26-s − 0.188·28-s + 1.11·29-s − 0.176·32-s + 0.338·35-s + 0.328·37-s − 0.648·38-s + 0.316·40-s + 1.56·41-s − 0.609·43-s + 0.603·44-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.478435454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.478435454\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.96183702402217, −14.57651454754093, −13.91280411111811, −13.33117344106646, −12.55465243444988, −12.19168809344693, −11.65846841415180, −11.29075322046001, −10.69162400525542, −10.03919162800853, −9.405071114042273, −9.139509196259878, −8.520968324733589, −7.814827011868384, −7.373523003332422, −6.900858844331408, −6.306468387582416, −5.671612481638654, −4.707412295718645, −4.354007191754104, −3.255256693284660, −3.192085028421308, −2.118513263802338, −1.136697482168727, −0.5961874363462464,
0.5961874363462464, 1.136697482168727, 2.118513263802338, 3.192085028421308, 3.255256693284660, 4.354007191754104, 4.707412295718645, 5.671612481638654, 6.306468387582416, 6.900858844331408, 7.373523003332422, 7.814827011868384, 8.520968324733589, 9.139509196259878, 9.405071114042273, 10.03919162800853, 10.69162400525542, 11.29075322046001, 11.65846841415180, 12.19168809344693, 12.55465243444988, 13.33117344106646, 13.91280411111811, 14.57651454754093, 14.96183702402217
