| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s − 11-s − 12-s − 2·15-s + 16-s − 17-s − 18-s − 8·19-s + 2·20-s + 22-s − 4·23-s + 24-s − 25-s − 27-s + 6·29-s + 2·30-s + 4·31-s − 32-s + 33-s + 34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.301·11-s − 0.288·12-s − 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.83·19-s + 0.447·20-s + 0.213·22-s − 0.834·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.365·30-s + 0.718·31-s − 0.176·32-s + 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.22266843533780, −12.87602674475962, −12.32830466732279, −11.96320959931309, −11.31144359093437, −10.86802073227257, −10.46548134704008, −10.14759144316686, −9.541054425948338, −9.249941775034225, −8.600162062330122, −8.072466179097190, −7.757503360103789, −7.033565367820357, −6.391138157768567, −6.134107166873865, −5.896969272709818, −5.052459092935579, −4.444970973122485, −4.156105058849367, −3.162091840588813, −2.471821744954716, −2.135102883255655, −1.475904962105135, −0.7292138592155097, 0,
0.7292138592155097, 1.475904962105135, 2.135102883255655, 2.471821744954716, 3.162091840588813, 4.156105058849367, 4.444970973122485, 5.052459092935579, 5.896969272709818, 6.134107166873865, 6.391138157768567, 7.033565367820357, 7.757503360103789, 8.072466179097190, 8.600162062330122, 9.249941775034225, 9.541054425948338, 10.14759144316686, 10.46548134704008, 10.86802073227257, 11.31144359093437, 11.96320959931309, 12.32830466732279, 12.87602674475962, 13.22266843533780
