L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s + 13-s + 4·14-s + 16-s + 3·17-s − 4·19-s + 26-s + 4·28-s + 9·29-s − 4·31-s + 32-s + 3·34-s + 37-s − 4·38-s + 6·41-s − 8·43-s + 12·47-s + 9·49-s + 52-s + 6·53-s + 4·56-s + 9·58-s − 61-s − 4·62-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.727·17-s − 0.917·19-s + 0.196·26-s + 0.755·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s + 0.514·34-s + 0.164·37-s − 0.648·38-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 9/7·49-s + 0.138·52-s + 0.824·53-s + 0.534·56-s + 1.18·58-s − 0.128·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.931403523\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.931403523\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−8.379379958879117511596030659261, −7.66569984112065354881594944845, −6.99135217510543484317355284922, −6.02370391570014613245592694812, −5.41898743106309665406075599896, −4.58419669367352936522826817013, −4.10307283039977795364198475320, −2.97037729574354355125627680754, −2.03536070446975091615350100174, −1.11218324936015124688879746682,
1.11218324936015124688879746682, 2.03536070446975091615350100174, 2.97037729574354355125627680754, 4.10307283039977795364198475320, 4.58419669367352936522826817013, 5.41898743106309665406075599896, 6.02370391570014613245592694812, 6.99135217510543484317355284922, 7.66569984112065354881594944845, 8.379379958879117511596030659261