L(s) = 1 | + 2-s + 0.732·3-s + 4-s − 5-s + 0.732·6-s − 3·7-s + 8-s − 2.46·9-s − 10-s + 3·11-s + 0.732·12-s − 3·14-s − 0.732·15-s + 16-s − 8.19·17-s − 2.46·18-s − 0.464·19-s − 20-s − 2.19·21-s + 3·22-s − 9.46·23-s + 0.732·24-s + 25-s − 4·27-s − 3·28-s − 2.53·29-s − 0.732·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.422·3-s + 0.5·4-s − 0.447·5-s + 0.298·6-s − 1.13·7-s + 0.353·8-s − 0.821·9-s − 0.316·10-s + 0.904·11-s + 0.211·12-s − 0.801·14-s − 0.189·15-s + 0.250·16-s − 1.98·17-s − 0.580·18-s − 0.106·19-s − 0.223·20-s − 0.479·21-s + 0.639·22-s − 1.97·23-s + 0.149·24-s + 0.200·25-s − 0.769·27-s − 0.566·28-s − 0.470·29-s − 0.133·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 - 0.732T + 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 17 | \( 1 + 8.19T + 17T^{2} \) |
| 19 | \( 1 + 0.464T + 19T^{2} \) |
| 23 | \( 1 + 9.46T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 + 0.803T + 37T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 + 3T + 47T^{2} \) |
| 53 | \( 1 - 0.464T + 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 6.19T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 4.19T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 - 6.80T + 89T^{2} \) |
| 97 | \( 1 - 9.12T + 97T^{2} \) |
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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
−8.944716277386641422487936105854, −8.199516084603582612628966258391, −7.20605978157107403303586388096, −6.30858073873753054875890997751, −5.98495258915817000684050056617, −4.48887764464873167908383063215, −3.90306066672733043556298014021, −3.00422847472893194405684055146, −2.09013708115895749655144303646, 0,
2.09013708115895749655144303646, 3.00422847472893194405684055146, 3.90306066672733043556298014021, 4.48887764464873167908383063215, 5.98495258915817000684050056617, 6.30858073873753054875890997751, 7.20605978157107403303586388096, 8.199516084603582612628966258391, 8.944716277386641422487936105854