L(s) = 1 | + 3.22·2-s + 8.71·3-s + 2.37·4-s − 13.5·5-s + 28.0·6-s + 24.0·7-s − 18.1·8-s + 49.0·9-s − 43.5·10-s − 71.6·11-s + 20.6·12-s + 26.7·13-s + 77.5·14-s − 117.·15-s − 77.3·16-s + 35.6·17-s + 157.·18-s − 88.4·19-s − 32.0·20-s + 209.·21-s − 230.·22-s − 64.6·23-s − 158.·24-s + 57.6·25-s + 86.2·26-s + 192.·27-s + 57.1·28-s + ⋯ |
L(s) = 1 | + 1.13·2-s + 1.67·3-s + 0.296·4-s − 1.20·5-s + 1.91·6-s + 1.29·7-s − 0.800·8-s + 1.81·9-s − 1.37·10-s − 1.96·11-s + 0.497·12-s + 0.571·13-s + 1.47·14-s − 2.02·15-s − 1.20·16-s + 0.508·17-s + 2.06·18-s − 1.06·19-s − 0.358·20-s + 2.18·21-s − 2.23·22-s − 0.586·23-s − 1.34·24-s + 0.461·25-s + 0.650·26-s + 1.36·27-s + 0.385·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 - 3.22T + 8T^{2} \) |
| 3 | \( 1 - 8.71T + 27T^{2} \) |
| 5 | \( 1 + 13.5T + 125T^{2} \) |
| 7 | \( 1 - 24.0T + 343T^{2} \) |
| 11 | \( 1 + 71.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 26.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 35.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 64.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 106.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 7.27T + 2.97e4T^{2} \) |
| 37 | \( 1 - 195.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 404.T + 6.89e4T^{2} \) |
| 47 | \( 1 - 31.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 34.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 486.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 491.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 102.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 119.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 123.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 34.0T + 4.93e5T^{2} \) |
| 83 | \( 1 + 681.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.56e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 93.1T + 9.12e5T^{2} \) |
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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
−8.134790394892664721668084010630, −8.051721042882407720896301666926, −7.24143045870331519378061960324, −5.79993886166207043274328224484, −4.82623022721079908541928793993, −4.26149329520750580263781310869, −3.50729102261652822371173611896, −2.76355388019066949805908595814, −1.85344253317941680055920182006, 0,
1.85344253317941680055920182006, 2.76355388019066949805908595814, 3.50729102261652822371173611896, 4.26149329520750580263781310869, 4.82623022721079908541928793993, 5.79993886166207043274328224484, 7.24143045870331519378061960324, 8.051721042882407720896301666926, 8.134790394892664721668084010630