| L(s) = 1 | + 1.36·3-s − 3.28·7-s − 1.13·9-s + 3.49·11-s + 3.41·13-s − 7.46·17-s + 3.49·19-s − 4.48·21-s + 23-s − 5.64·27-s − 3.46·29-s + 2.01·31-s + 4.77·33-s + 0.511·37-s + 4.66·39-s − 7.07·41-s − 2.76·43-s − 0.889·47-s + 3.76·49-s − 10.1·51-s − 14.2·53-s + 4.77·57-s − 4.71·59-s + 13.4·61-s + 3.71·63-s − 2.30·67-s + 1.36·69-s + ⋯ |
| L(s) = 1 | + 0.788·3-s − 1.24·7-s − 0.377·9-s + 1.05·11-s + 0.946·13-s − 1.80·17-s + 0.802·19-s − 0.978·21-s + 0.208·23-s − 1.08·27-s − 0.643·29-s + 0.361·31-s + 0.831·33-s + 0.0840·37-s + 0.746·39-s − 1.10·41-s − 0.421·43-s − 0.129·47-s + 0.537·49-s − 1.42·51-s − 1.95·53-s + 0.632·57-s − 0.613·59-s + 1.72·61-s + 0.468·63-s − 0.281·67-s + 0.164·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 1.36T + 3T^{2} \) |
| 7 | \( 1 + 3.28T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 7.46T + 17T^{2} \) |
| 19 | \( 1 - 3.49T + 19T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 - 2.01T + 31T^{2} \) |
| 37 | \( 1 - 0.511T + 37T^{2} \) |
| 41 | \( 1 + 7.07T + 41T^{2} \) |
| 43 | \( 1 + 2.76T + 43T^{2} \) |
| 47 | \( 1 + 0.889T + 47T^{2} \) |
| 53 | \( 1 + 14.2T + 53T^{2} \) |
| 59 | \( 1 + 4.71T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 2.30T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 - 3.70T + 73T^{2} \) |
| 79 | \( 1 + 7.97T + 79T^{2} \) |
| 83 | \( 1 - 9.42T + 83T^{2} \) |
| 89 | \( 1 - 0.801T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.147445700184492262444871974923, −7.13966345542285764936281232709, −6.46745848775082129803443314289, −6.06499290609694168657397738106, −4.90857028334821358683371695760, −3.84185520472874140010348793103, −3.41268378351922524326574516864, −2.58542720414934539616655177691, −1.51330792393091999993441701124, 0,
1.51330792393091999993441701124, 2.58542720414934539616655177691, 3.41268378351922524326574516864, 3.84185520472874140010348793103, 4.90857028334821358683371695760, 6.06499290609694168657397738106, 6.46745848775082129803443314289, 7.13966345542285764936281232709, 8.147445700184492262444871974923
