| L(s) = 1 | + 3-s − 3·7-s + 9-s + 5·13-s − 5·19-s − 3·21-s + 4·23-s + 27-s + 4·29-s + 5·31-s + 10·37-s + 5·39-s − 10·41-s + 43-s − 2·47-s + 2·49-s + 10·53-s − 5·57-s + 10·59-s − 5·61-s − 3·63-s − 3·67-s + 4·69-s + 10·71-s + 10·73-s + 81-s − 14·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.13·7-s + 1/3·9-s + 1.38·13-s − 1.14·19-s − 0.654·21-s + 0.834·23-s + 0.192·27-s + 0.742·29-s + 0.898·31-s + 1.64·37-s + 0.800·39-s − 1.56·41-s + 0.152·43-s − 0.291·47-s + 2/7·49-s + 1.37·53-s − 0.662·57-s + 1.30·59-s − 0.640·61-s − 0.377·63-s − 0.366·67-s + 0.481·69-s + 1.18·71-s + 1.17·73-s + 1/9·81-s − 1.53·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.051371601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.051371601\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−8.689212592593958564845829112530, −8.546766419499039931046050760157, −7.42986587304468267121837621140, −6.43613238042361999436017285803, −6.23051323416521086406521640418, −4.89501508930044077505548068755, −3.91510028078134490261793264148, −3.24284763757461778574670630879, −2.32095892195375385017423267081, −0.907770031859715511871219524535,
0.907770031859715511871219524535, 2.32095892195375385017423267081, 3.24284763757461778574670630879, 3.91510028078134490261793264148, 4.89501508930044077505548068755, 6.23051323416521086406521640418, 6.43613238042361999436017285803, 7.42986587304468267121837621140, 8.546766419499039931046050760157, 8.689212592593958564845829112530
