L(s) = 1 | − 18.0·2-s − 64.0·3-s + 198.·4-s + 13.5·5-s + 1.15e3·6-s + 343·7-s − 1.26e3·8-s + 1.92e3·9-s − 245.·10-s + 6.48e3·11-s − 1.26e4·12-s + 2.19e3·13-s − 6.19e3·14-s − 870.·15-s − 2.48e3·16-s − 3.10e4·17-s − 3.46e4·18-s − 6.93e3·19-s + 2.69e3·20-s − 2.19e4·21-s − 1.17e5·22-s + 6.15e3·23-s + 8.11e4·24-s − 7.79e4·25-s − 3.96e4·26-s + 1.70e4·27-s + 6.79e4·28-s + ⋯ |
L(s) = 1 | − 1.59·2-s − 1.37·3-s + 1.54·4-s + 0.0486·5-s + 2.18·6-s + 0.377·7-s − 0.874·8-s + 0.878·9-s − 0.0776·10-s + 1.46·11-s − 2.12·12-s + 0.277·13-s − 0.603·14-s − 0.0666·15-s − 0.151·16-s − 1.53·17-s − 1.40·18-s − 0.232·19-s + 0.0752·20-s − 0.517·21-s − 2.34·22-s + 0.105·23-s + 1.19·24-s − 0.997·25-s − 0.442·26-s + 0.166·27-s + 0.585·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.4520873913\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4520873913\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
good | 2 | \( 1 + 18.0T + 128T^{2} \) |
| 3 | \( 1 + 64.0T + 2.18e3T^{2} \) |
| 5 | \( 1 - 13.5T + 7.81e4T^{2} \) |
| 11 | \( 1 - 6.48e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 3.10e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 6.93e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.15e3T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.69e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.82e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.32e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.12e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 6.03e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.17e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.08e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.41e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.76e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 3.17e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 1.13e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.75e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.25e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.20e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.22e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 4.76e6T + 8.07e13T^{2} \) |
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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
−11.94319828250332100323343096885, −11.37605841175819778741518701794, −10.55871460256726442498553098432, −9.396307567946603511348804922010, −8.421625712703037553716533885800, −6.92950969265536916292549626966, −6.17565444265775355024513126867, −4.44781329698914411746320147937, −1.77316910285534149119624601364, −0.59501242932623183041398140908,
0.59501242932623183041398140908, 1.77316910285534149119624601364, 4.44781329698914411746320147937, 6.17565444265775355024513126867, 6.92950969265536916292549626966, 8.421625712703037553716533885800, 9.396307567946603511348804922010, 10.55871460256726442498553098432, 11.37605841175819778741518701794, 11.94319828250332100323343096885