| L(s) = 1 | − 3-s − 3·5-s − 7-s + 9-s + 5·13-s + 3·15-s − 8·19-s + 21-s + 23-s + 4·25-s − 27-s + 3·29-s − 2·31-s + 3·35-s − 7·37-s − 5·39-s + 9·41-s + 43-s − 3·45-s + 3·47-s + 49-s − 12·53-s + 8·57-s + 6·59-s + 14·61-s − 63-s − 15·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s − 0.377·7-s + 1/3·9-s + 1.38·13-s + 0.774·15-s − 1.83·19-s + 0.218·21-s + 0.208·23-s + 4/5·25-s − 0.192·27-s + 0.557·29-s − 0.359·31-s + 0.507·35-s − 1.15·37-s − 0.800·39-s + 1.40·41-s + 0.152·43-s − 0.447·45-s + 0.437·47-s + 1/7·49-s − 1.64·53-s + 1.05·57-s + 0.781·59-s + 1.79·61-s − 0.125·63-s − 1.86·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.52155684206561073341645667168, −6.67518484569444553793900631957, −6.32249153618949543978403158493, −5.44869862979028810984163064729, −4.51488140989069929467551471176, −3.93574107593318254849578479217, −3.42066663020126256699565937539, −2.22585205707958713903209715483, −0.979894419021066435174737834516, 0,
0.979894419021066435174737834516, 2.22585205707958713903209715483, 3.42066663020126256699565937539, 3.93574107593318254849578479217, 4.51488140989069929467551471176, 5.44869862979028810984163064729, 6.32249153618949543978403158493, 6.67518484569444553793900631957, 7.52155684206561073341645667168
