L(s) = 1 | + 1.81·3-s − 5-s − 1.81·7-s + 0.289·9-s + 1.28·11-s + 3.62·13-s − 1.81·15-s − 6.20·17-s − 3.10·19-s − 3.28·21-s + 1.42·23-s + 25-s − 4.91·27-s − 2·29-s − 4.15·31-s + 2.33·33-s + 1.81·35-s − 37-s + 6.57·39-s − 8.33·41-s − 2.57·43-s − 0.289·45-s − 1.81·47-s − 3.71·49-s − 11.2·51-s + 1.49·53-s − 1.28·55-s + ⋯ |
L(s) = 1 | + 1.04·3-s − 0.447·5-s − 0.685·7-s + 0.0963·9-s + 0.388·11-s + 1.00·13-s − 0.468·15-s − 1.50·17-s − 0.711·19-s − 0.717·21-s + 0.296·23-s + 0.200·25-s − 0.946·27-s − 0.371·29-s − 0.745·31-s + 0.407·33-s + 0.306·35-s − 0.164·37-s + 1.05·39-s − 1.30·41-s − 0.393·43-s − 0.0431·45-s − 0.264·47-s − 0.530·49-s − 1.57·51-s + 0.205·53-s − 0.173·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - 1.81T + 3T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 - 3.62T + 13T^{2} \) |
| 17 | \( 1 + 6.20T + 17T^{2} \) |
| 19 | \( 1 + 3.10T + 19T^{2} \) |
| 23 | \( 1 - 1.42T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 4.15T + 31T^{2} \) |
| 41 | \( 1 + 8.33T + 41T^{2} \) |
| 43 | \( 1 + 2.57T + 43T^{2} \) |
| 47 | \( 1 + 1.81T + 47T^{2} \) |
| 53 | \( 1 - 1.49T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 + 3.42T + 61T^{2} \) |
| 67 | \( 1 + 0.897T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 + 2.67T + 97T^{2} \) |
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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
−8.633412824873747205021043748091, −7.77960456587061629367694766245, −6.79124222380873456136563123936, −6.37250531845660969555041492810, −5.22903957158759112072751323507, −4.04772916843161779956105795228, −3.61577829212724707584007828636, −2.70399375343960977381559089105, −1.72812350424980092281370473415, 0,
1.72812350424980092281370473415, 2.70399375343960977381559089105, 3.61577829212724707584007828636, 4.04772916843161779956105795228, 5.22903957158759112072751323507, 6.37250531845660969555041492810, 6.79124222380873456136563123936, 7.77960456587061629367694766245, 8.633412824873747205021043748091