L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 5·7-s − 8-s + 9-s − 10-s + 11-s − 12-s + 6·13-s + 5·14-s − 15-s + 16-s + 4·17-s − 18-s + 20-s + 5·21-s − 22-s + 7·23-s + 24-s + 25-s − 6·26-s − 27-s − 5·28-s + 6·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.88·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 1.66·13-s + 1.33·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.223·20-s + 1.09·21-s − 0.213·22-s + 1.45·23-s + 0.204·24-s + 1/5·25-s − 1.17·26-s − 0.192·27-s − 0.944·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.320878186\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.320878186\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−16.55259624116890, −16.18671175582212, −15.60395350579221, −15.10657874160144, −14.15192046610076, −13.50584766572207, −12.97648293231160, −12.58095293922750, −11.82664051350225, −11.22166547442214, −10.46752979082809, −10.19563776814222, −9.411798919632051, −9.096495047997554, −8.372329239193445, −7.485764186712259, −6.670626885092130, −6.385961039225069, −5.892375323147863, −5.098226646371983, −3.896573123325929, −3.299989283584259, −2.632580200596660, −1.260495023524845, −0.7147988577571983,
0.7147988577571983, 1.260495023524845, 2.632580200596660, 3.299989283584259, 3.896573123325929, 5.098226646371983, 5.892375323147863, 6.385961039225069, 6.670626885092130, 7.485764186712259, 8.372329239193445, 9.096495047997554, 9.411798919632051, 10.19563776814222, 10.46752979082809, 11.22166547442214, 11.82664051350225, 12.58095293922750, 12.97648293231160, 13.50584766572207, 14.15192046610076, 15.10657874160144, 15.60395350579221, 16.18671175582212, 16.55259624116890