L(s) = 1 | − 2-s − 3·3-s − 4-s + 5-s + 3·6-s − 3·7-s + 3·8-s + 6·9-s − 10-s + 3·12-s + 4·13-s + 3·14-s − 3·15-s − 16-s − 6·18-s + 4·19-s − 20-s + 9·21-s − 8·23-s − 9·24-s + 25-s − 4·26-s − 9·27-s + 3·28-s + 6·29-s + 3·30-s − 2·31-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s + 0.447·5-s + 1.22·6-s − 1.13·7-s + 1.06·8-s + 2·9-s − 0.316·10-s + 0.866·12-s + 1.10·13-s + 0.801·14-s − 0.774·15-s − 1/4·16-s − 1.41·18-s + 0.917·19-s − 0.223·20-s + 1.96·21-s − 1.66·23-s − 1.83·24-s + 1/5·25-s − 0.784·26-s − 1.73·27-s + 0.566·28-s + 1.11·29-s + 0.547·30-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - T + p T^{2} \) |
| 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
−10.23018872792594256269541712332, −9.639657197632512206761948111905, −8.662767229663167523496604004309, −7.41714756127606912641079580252, −6.39601887363543932679446149965, −5.83442423326813044858896824051, −4.82796990935721592135174547060, −3.66174681926932165983830279437, −1.34412844528491965270904028569, 0,
1.34412844528491965270904028569, 3.66174681926932165983830279437, 4.82796990935721592135174547060, 5.83442423326813044858896824051, 6.39601887363543932679446149965, 7.41714756127606912641079580252, 8.662767229663167523496604004309, 9.639657197632512206761948111905, 10.23018872792594256269541712332