| L(s) = 1 | + 2.74·3-s − 1.42·7-s + 4.51·9-s + 0.375·11-s − 6.69·13-s − 1.44·17-s − 6.63·19-s − 3.90·21-s − 6.67·23-s + 4.16·27-s + 0.174·29-s + 7.61·31-s + 1.02·33-s − 8.13·37-s − 18.3·39-s + 7.97·41-s + 4.98·43-s − 1.11·47-s − 4.97·49-s − 3.95·51-s − 1.52·53-s − 18.1·57-s − 10.1·59-s − 8.96·61-s − 6.42·63-s − 8.03·67-s − 18.3·69-s + ⋯ |
| L(s) = 1 | + 1.58·3-s − 0.537·7-s + 1.50·9-s + 0.113·11-s − 1.85·13-s − 0.350·17-s − 1.52·19-s − 0.851·21-s − 1.39·23-s + 0.801·27-s + 0.0324·29-s + 1.36·31-s + 0.179·33-s − 1.33·37-s − 2.93·39-s + 1.24·41-s + 0.759·43-s − 0.162·47-s − 0.710·49-s − 0.554·51-s − 0.209·53-s − 2.40·57-s − 1.31·59-s − 1.14·61-s − 0.809·63-s − 0.981·67-s − 2.20·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 2.74T + 3T^{2} \) |
| 7 | \( 1 + 1.42T + 7T^{2} \) |
| 11 | \( 1 - 0.375T + 11T^{2} \) |
| 13 | \( 1 + 6.69T + 13T^{2} \) |
| 17 | \( 1 + 1.44T + 17T^{2} \) |
| 19 | \( 1 + 6.63T + 19T^{2} \) |
| 23 | \( 1 + 6.67T + 23T^{2} \) |
| 29 | \( 1 - 0.174T + 29T^{2} \) |
| 31 | \( 1 - 7.61T + 31T^{2} \) |
| 37 | \( 1 + 8.13T + 37T^{2} \) |
| 41 | \( 1 - 7.97T + 41T^{2} \) |
| 43 | \( 1 - 4.98T + 43T^{2} \) |
| 47 | \( 1 + 1.11T + 47T^{2} \) |
| 53 | \( 1 + 1.52T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 + 8.96T + 61T^{2} \) |
| 67 | \( 1 + 8.03T + 67T^{2} \) |
| 71 | \( 1 + 7.84T + 71T^{2} \) |
| 73 | \( 1 - 8.05T + 73T^{2} \) |
| 79 | \( 1 + 0.231T + 79T^{2} \) |
| 83 | \( 1 - 7.21T + 83T^{2} \) |
| 89 | \( 1 - 5.85T + 89T^{2} \) |
| 97 | \( 1 - 8.49T + 97T^{2} \) |
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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
−7.949933276330183384158370752132, −7.68222601298473413899519210205, −6.73303788798758049346968070385, −6.07390573726651803994177292225, −4.71175231371712244023391389531, −4.22141398161134357029645812897, −3.22899582593578292900470707375, −2.48377480319880281784708023715, −1.91611783894612412282834198875, 0,
1.91611783894612412282834198875, 2.48377480319880281784708023715, 3.22899582593578292900470707375, 4.22141398161134357029645812897, 4.71175231371712244023391389531, 6.07390573726651803994177292225, 6.73303788798758049346968070385, 7.68222601298473413899519210205, 7.949933276330183384158370752132
