L(s) = 1 | + 2-s − 3-s − 4-s − 2·5-s − 6-s − 3·8-s + 9-s − 2·10-s + 12-s − 13-s + 2·15-s − 16-s − 17-s + 18-s + 4·19-s + 2·20-s + 3·24-s − 25-s − 26-s − 27-s + 2·29-s + 2·30-s − 8·31-s + 5·32-s − 34-s − 36-s − 2·37-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s − 0.408·6-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.288·12-s − 0.277·13-s + 0.516·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s + 0.612·24-s − 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.371·29-s + 0.365·30-s − 1.43·31-s + 0.883·32-s − 0.171·34-s − 1/6·36-s − 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80223 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80223 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5974916754\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5974916754\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−13.85213337040087, −13.64311191551793, −12.85432306327749, −12.34591112134823, −12.29859911365268, −11.54576100915024, −11.20039645897983, −10.67546900803568, −9.969566104722212, −9.413067592558907, −9.067464235564383, −8.368636031462496, −7.729004013825360, −7.434109626540978, −6.674124285548644, −6.153506639609774, −5.517239433724197, −5.088898260736896, −4.565066131373711, −4.018865178446543, −3.496052280193142, −3.003750485889008, −2.073187345113229, −1.134986205714987, −0.2597210392578932,
0.2597210392578932, 1.134986205714987, 2.073187345113229, 3.003750485889008, 3.496052280193142, 4.018865178446543, 4.565066131373711, 5.088898260736896, 5.517239433724197, 6.153506639609774, 6.674124285548644, 7.434109626540978, 7.729004013825360, 8.368636031462496, 9.067464235564383, 9.413067592558907, 9.969566104722212, 10.67546900803568, 11.20039645897983, 11.54576100915024, 12.29859911365268, 12.34591112134823, 12.85432306327749, 13.64311191551793, 13.85213337040087