| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s + 7-s − 8-s + 9-s − 2·11-s + 12-s − 14-s + 16-s − 17-s − 18-s + 19-s + 21-s + 2·22-s − 24-s + 27-s + 28-s + 29-s − 8·31-s − 32-s − 2·33-s + 34-s + 36-s − 2·37-s − 38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.229·19-s + 0.218·21-s + 0.426·22-s − 0.204·24-s + 0.192·27-s + 0.188·28-s + 0.185·29-s − 1.43·31-s − 0.176·32-s − 0.348·33-s + 0.171·34-s + 1/6·36-s − 0.328·37-s − 0.162·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−16.07133328723017, −15.64540188467604, −14.91080219477291, −14.65700458934424, −13.94919547081740, −13.26444554782037, −12.92159933046573, −12.07693133257961, −11.65783803378738, −10.77197189545815, −10.63122747732843, −9.755029189988889, −9.353895889858668, −8.662267212097445, −8.198914658558701, −7.655167555063417, −7.057767693571091, −6.496031160651279, −5.513256997040389, −5.091238831874072, −4.099596103361603, −3.453100937497644, −2.600892681013828, −2.009301501066238, −1.159900044175050, 0,
1.159900044175050, 2.009301501066238, 2.600892681013828, 3.453100937497644, 4.099596103361603, 5.091238831874072, 5.513256997040389, 6.496031160651279, 7.057767693571091, 7.655167555063417, 8.198914658558701, 8.662267212097445, 9.353895889858668, 9.755029189988889, 10.63122747732843, 10.77197189545815, 11.65783803378738, 12.07693133257961, 12.92159933046573, 13.26444554782037, 13.94919547081740, 14.65700458934424, 14.91080219477291, 15.64540188467604, 16.07133328723017
