L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 2·7-s + 8-s + 9-s + 2·10-s + 11-s − 12-s + 6·13-s + 2·14-s − 2·15-s + 16-s + 2·17-s + 18-s − 6·19-s + 2·20-s − 2·21-s + 22-s − 4·23-s − 24-s − 25-s + 6·26-s − 27-s + 2·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.288·12-s + 1.66·13-s + 0.534·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 1.37·19-s + 0.447·20-s − 0.436·21-s + 0.213·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 1.17·26-s − 0.192·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90354 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 37 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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
−14.00514081296993, −13.79714139021530, −13.02589492592811, −12.70494745934158, −12.30228954217364, −11.50669626632667, −11.19814731468894, −10.82121689900934, −10.32095090942594, −9.746899110556405, −9.153587679370276, −8.522484790796852, −8.063790949547041, −7.518785341273401, −6.575096054635973, −6.408660699329337, −5.876091412423056, −5.439523411598688, −4.879032223610121, −4.148007976338357, −3.830720599719429, −3.083516535715450, −2.128584902380906, −1.662205020801775, −1.248491298511818, 0,
1.248491298511818, 1.662205020801775, 2.128584902380906, 3.083516535715450, 3.830720599719429, 4.148007976338357, 4.879032223610121, 5.439523411598688, 5.876091412423056, 6.408660699329337, 6.575096054635973, 7.518785341273401, 8.063790949547041, 8.522484790796852, 9.153587679370276, 9.746899110556405, 10.32095090942594, 10.82121689900934, 11.19814731468894, 11.50669626632667, 12.30228954217364, 12.70494745934158, 13.02589492592811, 13.79714139021530, 14.00514081296993