L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s + 4·11-s − 12-s + 2·14-s + 15-s + 16-s − 18-s + 19-s − 20-s + 2·21-s − 4·22-s − 9·23-s + 24-s + 25-s − 27-s − 2·28-s − 10·29-s − 30-s − 5·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.436·21-s − 0.852·22-s − 1.87·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.377·28-s − 1.85·29-s − 0.182·30-s − 0.898·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.41572463141746, −13.82953198284379, −13.11121968784741, −12.72997077557069, −12.16757156604974, −11.66784591106059, −11.49468329527251, −10.85968730854072, −10.12892954850586, −9.989743551165794, −9.310206526212931, −8.883327721206588, −8.428609743246299, −7.538593122926655, −7.369190668305230, −6.725225820319696, −6.246746334043192, −5.741182871780383, −5.246880706896285, −4.288339804356522, −3.676221307634477, −3.581620064003082, −2.471046701471219, −1.762193678994549, −1.221544961359222, 0, 0,
1.221544961359222, 1.762193678994549, 2.471046701471219, 3.581620064003082, 3.676221307634477, 4.288339804356522, 5.246880706896285, 5.741182871780383, 6.246746334043192, 6.725225820319696, 7.369190668305230, 7.538593122926655, 8.428609743246299, 8.883327721206588, 9.310206526212931, 9.989743551165794, 10.12892954850586, 10.85968730854072, 11.49468329527251, 11.66784591106059, 12.16757156604974, 12.72997077557069, 13.11121968784741, 13.82953198284379, 14.41572463141746