L(s) = 1 | + 3-s − 5-s − 1.23·7-s − 2·9-s + 3.23·11-s − 6.23·13-s − 15-s + 2.47·17-s + 5.70·19-s − 1.23·21-s + 23-s + 25-s − 5·27-s − 0.527·29-s − 4.23·31-s + 3.23·33-s + 1.23·35-s − 9.70·37-s − 6.23·39-s − 7.47·41-s − 3.70·43-s + 2·45-s − 9.47·47-s − 5.47·49-s + 2.47·51-s − 6·53-s − 3.23·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.467·7-s − 0.666·9-s + 0.975·11-s − 1.72·13-s − 0.258·15-s + 0.599·17-s + 1.30·19-s − 0.269·21-s + 0.208·23-s + 0.200·25-s − 0.962·27-s − 0.0980·29-s − 0.760·31-s + 0.563·33-s + 0.208·35-s − 1.59·37-s − 0.998·39-s − 1.16·41-s − 0.565·43-s + 0.298·45-s − 1.38·47-s − 0.781·49-s + 0.346·51-s − 0.824·53-s − 0.436·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - T + 3T^{2} \) |
| 7 | \( 1 + 1.23T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 6.23T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 5.70T + 19T^{2} \) |
| 29 | \( 1 + 0.527T + 29T^{2} \) |
| 31 | \( 1 + 4.23T + 31T^{2} \) |
| 37 | \( 1 + 9.70T + 37T^{2} \) |
| 41 | \( 1 + 7.47T + 41T^{2} \) |
| 43 | \( 1 + 3.70T + 43T^{2} \) |
| 47 | \( 1 + 9.47T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 12.1T + 61T^{2} \) |
| 67 | \( 1 + 9.70T + 67T^{2} \) |
| 71 | \( 1 - 6.23T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 + 8.76T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 - 2.76T + 89T^{2} \) |
| 97 | \( 1 + 6.18T + 97T^{2} \) |
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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
−8.926218000403695387159332084639, −8.058178405358575091257182183553, −7.32807957413293584711682161824, −6.68710021124262154165815009113, −5.50440479705808773231151571596, −4.79450616879413474957441780601, −3.45180272862097588411624472908, −3.11583686011888288294257461115, −1.75398930023516601792799084058, 0,
1.75398930023516601792799084058, 3.11583686011888288294257461115, 3.45180272862097588411624472908, 4.79450616879413474957441780601, 5.50440479705808773231151571596, 6.68710021124262154165815009113, 7.32807957413293584711682161824, 8.058178405358575091257182183553, 8.926218000403695387159332084639