| L(s) = 1 | + 3-s + 5-s − 7-s + 9-s − 2·11-s − 4·13-s + 15-s − 6·17-s − 4·19-s − 21-s + 23-s + 25-s + 27-s − 2·29-s − 2·31-s − 2·33-s − 35-s − 4·37-s − 4·39-s − 2·41-s + 4·43-s + 45-s + 4·47-s + 49-s − 6·51-s + 12·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s + 1/3·9-s − 0.603·11-s − 1.10·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s + 0.208·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 0.359·31-s − 0.348·33-s − 0.169·35-s − 0.657·37-s − 0.640·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s + 0.583·47-s + 1/7·49-s − 0.840·51-s + 1.64·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 154560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−13.46664912535460, −13.07414753537692, −12.81852971437500, −12.20453752729767, −11.73539467062018, −11.03721027478874, −10.55159963843148, −10.26067349888187, −9.679675215244323, −9.140061172834774, −8.849017173175471, −8.312114225807463, −7.737008578164940, −7.059169774899163, −6.884728025935139, −6.283917767338426, −5.452172305991505, −5.266323220047130, −4.337327928189305, −4.174369188380622, −3.315616883484787, −2.671736159241698, −2.202122301404693, −1.910928632345342, −0.7382148283081022, 0,
0.7382148283081022, 1.910928632345342, 2.202122301404693, 2.671736159241698, 3.315616883484787, 4.174369188380622, 4.337327928189305, 5.266323220047130, 5.452172305991505, 6.283917767338426, 6.884728025935139, 7.059169774899163, 7.737008578164940, 8.312114225807463, 8.849017173175471, 9.140061172834774, 9.679675215244323, 10.26067349888187, 10.55159963843148, 11.03721027478874, 11.73539467062018, 12.20453752729767, 12.81852971437500, 13.07414753537692, 13.46664912535460
