L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s + 8-s − 2·9-s + 10-s + 11-s − 12-s + 13-s − 15-s + 16-s − 2·18-s − 2·19-s + 20-s + 22-s − 6·23-s − 24-s + 25-s + 26-s + 5·27-s − 9·29-s − 30-s + 4·31-s + 32-s − 33-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s − 0.471·18-s − 0.458·19-s + 0.223·20-s + 0.213·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.962·27-s − 1.67·29-s − 0.182·30-s + 0.718·31-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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
−7.71130651766922566939963786274, −6.78613813962154446666117135467, −6.25698537194541067572226644769, −5.61300996013221853386988995769, −5.09806924163523724202718911022, −4.11708386991979073274561102167, −3.41575162378396414612944744403, −2.39134820969446573750542719327, −1.53320715272253076634135659671, 0,
1.53320715272253076634135659671, 2.39134820969446573750542719327, 3.41575162378396414612944744403, 4.11708386991979073274561102167, 5.09806924163523724202718911022, 5.61300996013221853386988995769, 6.25698537194541067572226644769, 6.78613813962154446666117135467, 7.71130651766922566939963786274