| L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s − 11-s − 5·13-s + 2·15-s + 4·17-s − 6·19-s + 2·21-s − 23-s − 25-s − 27-s − 6·29-s − 4·31-s + 33-s + 4·35-s − 7·37-s + 5·39-s − 2·45-s + 8·47-s − 3·49-s − 4·51-s − 10·53-s + 2·55-s + 6·57-s − 5·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.38·13-s + 0.516·15-s + 0.970·17-s − 1.37·19-s + 0.436·21-s − 0.208·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.174·33-s + 0.676·35-s − 1.15·37-s + 0.800·39-s − 0.298·45-s + 1.16·47-s − 3/7·49-s − 0.560·51-s − 1.37·53-s + 0.269·55-s + 0.794·57-s − 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80688 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80688 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 41 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
| 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
−14.20651065552041, −13.81345395140039, −12.92561091422060, −12.65549530302379, −12.25525416871699, −11.95947157038627, −11.10950539201395, −10.90965090396268, −10.20495521057807, −9.831775479919969, −9.275737953234909, −8.732669956315176, −7.907068979336204, −7.559995910236130, −7.252194671619717, −6.359458474990120, −6.171451132117376, −5.218667795361617, −4.999457322060886, −4.186180400759595, −3.668412907891397, −3.182418480606444, −2.292029678291968, −1.732562254208948, −0.4964285356542630, 0,
0.4964285356542630, 1.732562254208948, 2.292029678291968, 3.182418480606444, 3.668412907891397, 4.186180400759595, 4.999457322060886, 5.218667795361617, 6.171451132117376, 6.359458474990120, 7.252194671619717, 7.559995910236130, 7.907068979336204, 8.732669956315176, 9.275737953234909, 9.831775479919969, 10.20495521057807, 10.90965090396268, 11.10950539201395, 11.95947157038627, 12.25525416871699, 12.65549530302379, 12.92561091422060, 13.81345395140039, 14.20651065552041
