| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 3·7-s + 8-s + 9-s − 10-s + 12-s − 2·13-s − 3·14-s − 15-s + 16-s + 5·17-s + 18-s − 5·19-s − 20-s − 3·21-s + 23-s + 24-s − 4·25-s − 2·26-s + 27-s − 3·28-s + 29-s − 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.288·12-s − 0.554·13-s − 0.801·14-s − 0.258·15-s + 1/4·16-s + 1.21·17-s + 0.235·18-s − 1.14·19-s − 0.223·20-s − 0.654·21-s + 0.208·23-s + 0.204·24-s − 4/5·25-s − 0.392·26-s + 0.192·27-s − 0.566·28-s + 0.185·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−7.923590315396760910210599695211, −7.30433032056213362975844401629, −6.63780146192769853438532842314, −5.85002215800991742729359386302, −5.02867488664786883125829221073, −4.02304213138004994505406446002, −3.48011828681751335152629581196, −2.77007878365070203716635022148, −1.71713613901716485833127746692, 0,
1.71713613901716485833127746692, 2.77007878365070203716635022148, 3.48011828681751335152629581196, 4.02304213138004994505406446002, 5.02867488664786883125829221073, 5.85002215800991742729359386302, 6.63780146192769853438532842314, 7.30433032056213362975844401629, 7.923590315396760910210599695211
