L(s) = 1 | + 2-s − 3-s + 4-s + 1.23·5-s − 6-s + 8-s + 9-s + 1.23·10-s − 5.23·11-s − 12-s + 1.23·13-s − 1.23·15-s + 16-s − 2.76·17-s + 18-s + 19-s + 1.23·20-s − 5.23·22-s + 5.23·23-s − 24-s − 3.47·25-s + 1.23·26-s − 27-s + 4.47·29-s − 1.23·30-s − 2.47·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.552·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.390·10-s − 1.57·11-s − 0.288·12-s + 0.342·13-s − 0.319·15-s + 0.250·16-s − 0.670·17-s + 0.235·18-s + 0.229·19-s + 0.276·20-s − 1.11·22-s + 1.09·23-s − 0.204·24-s − 0.694·25-s + 0.242·26-s − 0.192·27-s + 0.830·29-s − 0.225·30-s − 0.444·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.593166428\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.593166428\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 + 5.23T + 11T^{2} \) |
| 13 | \( 1 - 1.23T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 23 | \( 1 - 5.23T + 23T^{2} \) |
| 29 | \( 1 - 4.47T + 29T^{2} \) |
| 31 | \( 1 + 2.47T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 8.94T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 - 13.4T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 + 2.47T + 61T^{2} \) |
| 67 | \( 1 - 9.23T + 67T^{2} \) |
| 71 | \( 1 - 2.47T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 - 11.7T + 79T^{2} \) |
| 83 | \( 1 - 14.4T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 - 7.70T + 97T^{2} \) |
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
−7.86637903135272666746254309936, −7.40603679150419976923059556588, −6.40542502102260374149600246803, −5.99542058142335012496338269966, −5.14339485458142725946849086761, −4.78848796899712931112418250379, −3.76184255859979585466965420152, −2.75000793518615802458021486801, −2.10825231902412426102919481798, −0.78417074611072947142556009410,
0.78417074611072947142556009410, 2.10825231902412426102919481798, 2.75000793518615802458021486801, 3.76184255859979585466965420152, 4.78848796899712931112418250379, 5.14339485458142725946849086761, 5.99542058142335012496338269966, 6.40542502102260374149600246803, 7.40603679150419976923059556588, 7.86637903135272666746254309936