| L(s) = 1 | − 2-s − 3-s + 4-s + 4·5-s + 6-s − 4·7-s − 8-s + 9-s − 4·10-s − 12-s + 4·14-s − 4·15-s + 16-s − 6·17-s − 18-s + 19-s + 4·20-s + 4·21-s + 6·23-s + 24-s + 11·25-s − 27-s − 4·28-s − 6·29-s + 4·30-s + 10·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.78·5-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.26·10-s − 0.288·12-s + 1.06·14-s − 1.03·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.229·19-s + 0.894·20-s + 0.872·21-s + 1.25·23-s + 0.204·24-s + 11/5·25-s − 0.192·27-s − 0.755·28-s − 1.11·29-s + 0.730·30-s + 1.79·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| 113 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 12 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.71057692986472, −16.17062216510382, −15.59317270232023, −15.08259308484567, −14.19292412022844, −13.45882883913358, −13.21962386723922, −12.73292644138163, −12.00567540508149, −11.23111606754041, −10.51154304758590, −10.27649765417188, −9.599995787581816, −9.038728106102049, −8.924632692064198, −7.636579962842694, −6.728617772844334, −6.582336018165581, −6.050929062842574, −5.332607180305933, −4.649022605793482, −3.413360827621125, −2.706198156886974, −2.009209144097549, −1.095993255036032, 0,
1.095993255036032, 2.009209144097549, 2.706198156886974, 3.413360827621125, 4.649022605793482, 5.332607180305933, 6.050929062842574, 6.582336018165581, 6.728617772844334, 7.636579962842694, 8.924632692064198, 9.038728106102049, 9.599995787581816, 10.27649765417188, 10.51154304758590, 11.23111606754041, 12.00567540508149, 12.73292644138163, 13.21962386723922, 13.45882883913358, 14.19292412022844, 15.08259308484567, 15.59317270232023, 16.17062216510382, 16.71057692986472
