| L(s) = 1 | − 5-s − 3·7-s + 2·11-s + 13-s + 3·17-s − 2·19-s + 4·23-s − 4·25-s − 2·29-s − 4·31-s + 3·35-s + 5·37-s + 12·41-s − 7·43-s − 9·47-s + 2·49-s − 4·53-s − 2·55-s + 6·59-s − 4·61-s − 65-s + 10·67-s − 15·71-s − 2·73-s − 6·77-s + 8·79-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 0.603·11-s + 0.277·13-s + 0.727·17-s − 0.458·19-s + 0.834·23-s − 4/5·25-s − 0.371·29-s − 0.718·31-s + 0.507·35-s + 0.821·37-s + 1.87·41-s − 1.06·43-s − 1.31·47-s + 2/7·49-s − 0.549·53-s − 0.269·55-s + 0.781·59-s − 0.512·61-s − 0.124·65-s + 1.22·67-s − 1.78·71-s − 0.234·73-s − 0.683·77-s + 0.900·79-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.069287068407326044308270083618, −7.43349489084068026273091343648, −6.58077620798120174116715076060, −6.08062567625954929148194792858, −5.16574128381926105856746207319, −4.07091054811134105912777170722, −3.54020682240368361077395499750, −2.67039823935118854544464566270, −1.33035396953510359861485977690, 0,
1.33035396953510359861485977690, 2.67039823935118854544464566270, 3.54020682240368361077395499750, 4.07091054811134105912777170722, 5.16574128381926105856746207319, 6.08062567625954929148194792858, 6.58077620798120174116715076060, 7.43349489084068026273091343648, 8.069287068407326044308270083618
