| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 5.10·7-s − 8-s + 9-s + 10-s + 12-s + 1.33·13-s − 5.10·14-s − 15-s + 16-s + 0.775·17-s − 18-s − 0.0785·19-s − 20-s + 5.10·21-s + 6.64·23-s − 24-s + 25-s − 1.33·26-s + 27-s + 5.10·28-s − 2.01·29-s + 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s − 0.408·6-s + 1.93·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s + 0.288·12-s + 0.369·13-s − 1.36·14-s − 0.258·15-s + 0.250·16-s + 0.188·17-s − 0.235·18-s − 0.0180·19-s − 0.223·20-s + 1.11·21-s + 1.38·23-s − 0.204·24-s + 0.200·25-s − 0.261·26-s + 0.192·27-s + 0.965·28-s − 0.374·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.175142114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.175142114\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 5.10T + 7T^{2} \) |
| 13 | \( 1 - 1.33T + 13T^{2} \) |
| 17 | \( 1 - 0.775T + 17T^{2} \) |
| 19 | \( 1 + 0.0785T + 19T^{2} \) |
| 23 | \( 1 - 6.64T + 23T^{2} \) |
| 29 | \( 1 + 2.01T + 29T^{2} \) |
| 31 | \( 1 - 8.49T + 31T^{2} \) |
| 37 | \( 1 + 3.93T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 6.01T + 43T^{2} \) |
| 47 | \( 1 - 6.56T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 5.32T + 59T^{2} \) |
| 61 | \( 1 + 3.74T + 61T^{2} \) |
| 67 | \( 1 - 0.588T + 67T^{2} \) |
| 71 | \( 1 + 2.31T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + 9.67T + 89T^{2} \) |
| 97 | \( 1 + 13.4T + 97T^{2} \) |
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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
−8.552964951081933258389048096237, −7.984716809895484913474642056054, −7.34134796508125229967371494293, −6.65334866476654960228406454110, −5.36911321368511256568053467372, −4.77923635028222352722515884064, −3.84424762418073461148210403134, −2.82621497415763705522349213394, −1.80011881606315677455541622499, −1.02231059918694922923337638275,
1.02231059918694922923337638275, 1.80011881606315677455541622499, 2.82621497415763705522349213394, 3.84424762418073461148210403134, 4.77923635028222352722515884064, 5.36911321368511256568053467372, 6.65334866476654960228406454110, 7.34134796508125229967371494293, 7.984716809895484913474642056054, 8.552964951081933258389048096237
