L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 25·5-s + 36·6-s − 49·7-s − 64·8-s − 162·9-s − 100·10-s − 187·11-s − 144·12-s + 627·13-s + 196·14-s − 225·15-s + 256·16-s + 1.81e3·17-s + 648·18-s + 258·19-s + 400·20-s + 441·21-s + 748·22-s + 2.97e3·23-s + 576·24-s + 625·25-s − 2.50e3·26-s + 3.64e3·27-s − 784·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s − 2/3·9-s − 0.316·10-s − 0.465·11-s − 0.288·12-s + 1.02·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 1.52·17-s + 0.471·18-s + 0.163·19-s + 0.223·20-s + 0.218·21-s + 0.329·22-s + 1.17·23-s + 0.204·24-s + 1/5·25-s − 0.727·26-s + 0.962·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 70 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.017051004\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.017051004\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
good | 3 | \( 1 + p^{2} T + p^{5} T^{2} \) |
| 11 | \( 1 + 17 p T + p^{5} T^{2} \) |
| 13 | \( 1 - 627 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1813 T + p^{5} T^{2} \) |
| 19 | \( 1 - 258 T + p^{5} T^{2} \) |
| 23 | \( 1 - 2970 T + p^{5} T^{2} \) |
| 29 | \( 1 - 1299 T + p^{5} T^{2} \) |
| 31 | \( 1 - 1916 T + p^{5} T^{2} \) |
| 37 | \( 1 - 6578 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6676 T + p^{5} T^{2} \) |
| 43 | \( 1 - 3178 T + p^{5} T^{2} \) |
| 47 | \( 1 + 22001 T + p^{5} T^{2} \) |
| 53 | \( 1 - 26168 T + p^{5} T^{2} \) |
| 59 | \( 1 - 3932 T + p^{5} T^{2} \) |
| 61 | \( 1 + 48740 T + p^{5} T^{2} \) |
| 67 | \( 1 + 44832 T + p^{5} T^{2} \) |
| 71 | \( 1 - 63736 T + p^{5} T^{2} \) |
| 73 | \( 1 - 60470 T + p^{5} T^{2} \) |
| 79 | \( 1 + 43721 T + p^{5} T^{2} \) |
| 83 | \( 1 - 1172 p T + p^{5} T^{2} \) |
| 89 | \( 1 - 45560 T + p^{5} T^{2} \) |
| 97 | \( 1 + 57295 T + p^{5} T^{2} \) |
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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
−13.66622230802704197590456070706, −12.43080389466818216092553098836, −11.26871829003105819032180345463, −10.36202315317844147617136591421, −9.196018945458646807224067532598, −7.966202135995660929490627386416, −6.42376460321667965437252922449, −5.42684175191597055050200631682, −3.03091307706779167125837789824, −0.924719863083208071621048271525,
0.924719863083208071621048271525, 3.03091307706779167125837789824, 5.42684175191597055050200631682, 6.42376460321667965437252922449, 7.966202135995660929490627386416, 9.196018945458646807224067532598, 10.36202315317844147617136591421, 11.26871829003105819032180345463, 12.43080389466818216092553098836, 13.66622230802704197590456070706