L(s) = 1 | + 3·3-s − 5-s − 7-s + 6·9-s + 2·11-s − 13-s − 3·15-s − 3·17-s − 6·19-s − 3·21-s + 4·23-s − 4·25-s + 9·27-s + 2·29-s − 4·31-s + 6·33-s + 35-s + 3·37-s − 3·39-s + 5·43-s − 6·45-s − 13·47-s − 6·49-s − 9·51-s + 12·53-s − 2·55-s − 18·57-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.447·5-s − 0.377·7-s + 2·9-s + 0.603·11-s − 0.277·13-s − 0.774·15-s − 0.727·17-s − 1.37·19-s − 0.654·21-s + 0.834·23-s − 4/5·25-s + 1.73·27-s + 0.371·29-s − 0.718·31-s + 1.04·33-s + 0.169·35-s + 0.493·37-s − 0.480·39-s + 0.762·43-s − 0.894·45-s − 1.89·47-s − 6/7·49-s − 1.26·51-s + 1.64·53-s − 0.269·55-s − 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.804057193\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.804057193\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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
−12.76922791773221065882370025601, −11.44138555694141255918714539737, −10.16479571855116242915645828857, −9.175534225967753404519516979495, −8.542273196799616534200427297006, −7.53566579558648020530867420842, −6.53148501748885385532746785562, −4.43437896316507372499242030963, −3.47529908355398035375426875224, −2.17545984222807868215828295765,
2.17545984222807868215828295765, 3.47529908355398035375426875224, 4.43437896316507372499242030963, 6.53148501748885385532746785562, 7.53566579558648020530867420842, 8.542273196799616534200427297006, 9.175534225967753404519516979495, 10.16479571855116242915645828857, 11.44138555694141255918714539737, 12.76922791773221065882370025601