| L(s) = 1 | + 1.19·3-s + 1.19·7-s − 1.56·9-s − 5.86·11-s − 0.364·13-s + 1.19·17-s − 19-s + 1.43·21-s − 8.23·23-s − 5.46·27-s − 7.86·29-s + 7.30·31-s − 7.03·33-s + 7.13·37-s − 0.436·39-s + 2.43·41-s − 7.39·43-s − 13.7·47-s − 5.56·49-s + 1.43·51-s − 7.39·53-s − 1.19·57-s + 12.8·59-s − 1.30·61-s − 1.87·63-s + 11.9·67-s − 9.86·69-s + ⋯ |
| L(s) = 1 | + 0.692·3-s + 0.453·7-s − 0.521·9-s − 1.76·11-s − 0.101·13-s + 0.290·17-s − 0.229·19-s + 0.313·21-s − 1.71·23-s − 1.05·27-s − 1.46·29-s + 1.31·31-s − 1.22·33-s + 1.17·37-s − 0.0699·39-s + 0.380·41-s − 1.12·43-s − 1.99·47-s − 0.794·49-s + 0.201·51-s − 1.01·53-s − 0.158·57-s + 1.67·59-s − 0.166·61-s − 0.236·63-s + 1.45·67-s − 1.18·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 7 | \( 1 - 1.19T + 7T^{2} \) |
| 11 | \( 1 + 5.86T + 11T^{2} \) |
| 13 | \( 1 + 0.364T + 13T^{2} \) |
| 17 | \( 1 - 1.19T + 17T^{2} \) |
| 23 | \( 1 + 8.23T + 23T^{2} \) |
| 29 | \( 1 + 7.86T + 29T^{2} \) |
| 31 | \( 1 - 7.30T + 31T^{2} \) |
| 37 | \( 1 - 7.13T + 37T^{2} \) |
| 41 | \( 1 - 2.43T + 41T^{2} \) |
| 43 | \( 1 + 7.39T + 43T^{2} \) |
| 47 | \( 1 + 13.7T + 47T^{2} \) |
| 53 | \( 1 + 7.39T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 + 1.30T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 2.12T + 71T^{2} \) |
| 73 | \( 1 - 2.50T + 73T^{2} \) |
| 79 | \( 1 + 7.74T + 79T^{2} \) |
| 83 | \( 1 + 3.02T + 83T^{2} \) |
| 89 | \( 1 + 5.68T + 89T^{2} \) |
| 97 | \( 1 - 1.09T + 97T^{2} \) |
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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
−8.551461093160894423181193566992, −7.994359911729091262771557118885, −7.71812629623321670961146813273, −6.37252904337519794972227129130, −5.53337862469520074919136207588, −4.78877237873351340495902590237, −3.65467397697074916693274332468, −2.71064648842484936641733728989, −1.95437803332624454046645372862, 0,
1.95437803332624454046645372862, 2.71064648842484936641733728989, 3.65467397697074916693274332468, 4.78877237873351340495902590237, 5.53337862469520074919136207588, 6.37252904337519794972227129130, 7.71812629623321670961146813273, 7.994359911729091262771557118885, 8.551461093160894423181193566992
