L(s) = 1 | + 2.74·2-s + 3-s + 5.55·4-s − 5-s + 2.74·6-s + 9.78·8-s + 9-s − 2.74·10-s − 4.44·11-s + 5.55·12-s − 3.61·13-s − 15-s + 15.7·16-s + 4.94·17-s + 2.74·18-s − 2.74·19-s − 5.55·20-s − 12.2·22-s − 4.36·23-s + 9.78·24-s + 25-s − 9.94·26-s + 27-s + 0.660·29-s − 2.74·30-s − 1.25·31-s + 23.7·32-s + ⋯ |
L(s) = 1 | + 1.94·2-s + 0.577·3-s + 2.77·4-s − 0.447·5-s + 1.12·6-s + 3.45·8-s + 0.333·9-s − 0.869·10-s − 1.34·11-s + 1.60·12-s − 1.00·13-s − 0.258·15-s + 3.94·16-s + 1.19·17-s + 0.647·18-s − 0.629·19-s − 1.24·20-s − 2.60·22-s − 0.909·23-s + 1.99·24-s + 0.200·25-s − 1.95·26-s + 0.192·27-s + 0.122·29-s − 0.501·30-s − 0.225·31-s + 4.20·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.218876946\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.218876946\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 2.74T + 2T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 13 | \( 1 + 3.61T + 13T^{2} \) |
| 17 | \( 1 - 4.94T + 17T^{2} \) |
| 19 | \( 1 + 2.74T + 19T^{2} \) |
| 23 | \( 1 + 4.36T + 23T^{2} \) |
| 29 | \( 1 - 0.660T + 29T^{2} \) |
| 31 | \( 1 + 1.25T + 31T^{2} \) |
| 37 | \( 1 + 2.16T + 37T^{2} \) |
| 41 | \( 1 - 5.77T + 41T^{2} \) |
| 43 | \( 1 + 9.11T + 43T^{2} \) |
| 47 | \( 1 + 6.48T + 47T^{2} \) |
| 53 | \( 1 - 7.03T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 9.41T + 61T^{2} \) |
| 67 | \( 1 - 3.06T + 67T^{2} \) |
| 71 | \( 1 - 0.277T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 - 7.17T + 83T^{2} \) |
| 89 | \( 1 - 0.828T + 89T^{2} \) |
| 97 | \( 1 + 6.20T + 97T^{2} \) |
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
−10.53435024453582307162548142350, −9.955824491100674921804867294143, −8.140095036649554627926132711176, −7.62581096558099931924551834006, −6.74690463231542967314799229699, −5.56828641024196974311363271590, −4.89642009560994860244916979890, −3.90139661289689872939876853035, −2.99273505517766816283948025480, −2.12190859142618843532802223967,
2.12190859142618843532802223967, 2.99273505517766816283948025480, 3.90139661289689872939876853035, 4.89642009560994860244916979890, 5.56828641024196974311363271590, 6.74690463231542967314799229699, 7.62581096558099931924551834006, 8.140095036649554627926132711176, 9.955824491100674921804867294143, 10.53435024453582307162548142350