L(s) = 1 | − 3.15·3-s + 5-s − 0.579·7-s + 6.94·9-s − 0.683·11-s − 7.07·13-s − 3.15·15-s + 7.90·17-s + 0.469·19-s + 1.82·21-s − 3.47·23-s + 25-s − 12.4·27-s + 3.47·29-s + 6.32·31-s + 2.15·33-s − 0.579·35-s − 37-s + 22.2·39-s + 0.683·41-s − 11.5·43-s + 6.94·45-s + 5.00·47-s − 6.66·49-s − 24.9·51-s + 4.50·53-s − 0.683·55-s + ⋯ |
L(s) = 1 | − 1.82·3-s + 0.447·5-s − 0.219·7-s + 2.31·9-s − 0.206·11-s − 1.96·13-s − 0.814·15-s + 1.91·17-s + 0.107·19-s + 0.399·21-s − 0.724·23-s + 0.200·25-s − 2.39·27-s + 0.644·29-s + 1.13·31-s + 0.375·33-s − 0.0980·35-s − 0.164·37-s + 3.57·39-s + 0.106·41-s − 1.76·43-s + 1.03·45-s + 0.729·47-s − 0.951·49-s − 3.49·51-s + 0.619·53-s − 0.0921·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 + 3.15T + 3T^{2} \) |
| 7 | \( 1 + 0.579T + 7T^{2} \) |
| 11 | \( 1 + 0.683T + 11T^{2} \) |
| 13 | \( 1 + 7.07T + 13T^{2} \) |
| 17 | \( 1 - 7.90T + 17T^{2} \) |
| 19 | \( 1 - 0.469T + 19T^{2} \) |
| 23 | \( 1 + 3.47T + 23T^{2} \) |
| 29 | \( 1 - 3.47T + 29T^{2} \) |
| 31 | \( 1 - 6.32T + 31T^{2} \) |
| 41 | \( 1 - 0.683T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 5.00T + 47T^{2} \) |
| 53 | \( 1 - 4.50T + 53T^{2} \) |
| 59 | \( 1 + 6.85T + 59T^{2} \) |
| 61 | \( 1 - 3.71T + 61T^{2} \) |
| 67 | \( 1 - 7.53T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 + 8.76T + 79T^{2} \) |
| 83 | \( 1 - 1.84T + 83T^{2} \) |
| 89 | \( 1 + 4.74T + 89T^{2} \) |
| 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
−9.623765019657410152614826868058, −8.052867075013855094428823027770, −7.24769967871095655984626917852, −6.54302583208321796026004523629, −5.63926195334762692964517154073, −5.17767670735524238624078886764, −4.36366221468617630689058322095, −2.85353195642938557724018328330, −1.36441964615009145333558586742, 0,
1.36441964615009145333558586742, 2.85353195642938557724018328330, 4.36366221468617630689058322095, 5.17767670735524238624078886764, 5.63926195334762692964517154073, 6.54302583208321796026004523629, 7.24769967871095655984626917852, 8.052867075013855094428823027770, 9.623765019657410152614826868058