| L(s) = 1 | − 3.09·3-s − 0.0237·7-s + 6.56·9-s − 3.58·11-s + 3.77·13-s − 3.62·17-s + 2.43·19-s + 0.0735·21-s − 1.71·23-s − 11.0·27-s + 3.85·29-s + 6.00·31-s + 11.0·33-s − 0.369·37-s − 11.6·39-s − 7.80·41-s − 0.174·43-s − 7.81·47-s − 6.99·49-s + 11.2·51-s − 8.97·53-s − 7.51·57-s + 4.45·59-s + 9.21·61-s − 0.156·63-s + 4.47·67-s + 5.30·69-s + ⋯ |
| L(s) = 1 | − 1.78·3-s − 0.00899·7-s + 2.18·9-s − 1.08·11-s + 1.04·13-s − 0.878·17-s + 0.557·19-s + 0.0160·21-s − 0.357·23-s − 2.12·27-s + 0.716·29-s + 1.07·31-s + 1.92·33-s − 0.0607·37-s − 1.87·39-s − 1.21·41-s − 0.0266·43-s − 1.13·47-s − 0.999·49-s + 1.56·51-s − 1.23·53-s − 0.995·57-s + 0.580·59-s + 1.17·61-s − 0.0196·63-s + 0.546·67-s + 0.638·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3.09T + 3T^{2} \) |
| 7 | \( 1 + 0.0237T + 7T^{2} \) |
| 11 | \( 1 + 3.58T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 3.62T + 17T^{2} \) |
| 19 | \( 1 - 2.43T + 19T^{2} \) |
| 23 | \( 1 + 1.71T + 23T^{2} \) |
| 29 | \( 1 - 3.85T + 29T^{2} \) |
| 31 | \( 1 - 6.00T + 31T^{2} \) |
| 37 | \( 1 + 0.369T + 37T^{2} \) |
| 41 | \( 1 + 7.80T + 41T^{2} \) |
| 43 | \( 1 + 0.174T + 43T^{2} \) |
| 47 | \( 1 + 7.81T + 47T^{2} \) |
| 53 | \( 1 + 8.97T + 53T^{2} \) |
| 59 | \( 1 - 4.45T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 - 9.69T + 71T^{2} \) |
| 73 | \( 1 + 3.95T + 73T^{2} \) |
| 79 | \( 1 - 9.68T + 79T^{2} \) |
| 83 | \( 1 - 8.95T + 83T^{2} \) |
| 89 | \( 1 + 17.0T + 89T^{2} \) |
| 97 | \( 1 - 2.76T + 97T^{2} \) |
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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
−6.95599128022233037999837835873, −6.55080516200169462389383813282, −6.00447823280487635437260759004, −5.17852358776347635763205699770, −4.88038694548145246881297228023, −4.03257898473618359292068978633, −3.08256962119594564584818801008, −1.90819365906734675279550968191, −0.947414603582549709778159929173, 0,
0.947414603582549709778159929173, 1.90819365906734675279550968191, 3.08256962119594564584818801008, 4.03257898473618359292068978633, 4.88038694548145246881297228023, 5.17852358776347635763205699770, 6.00447823280487635437260759004, 6.55080516200169462389383813282, 6.95599128022233037999837835873
