| L(s) = 1 | + 5-s + 11-s − 13-s − 2·17-s + 4·19-s + 25-s + 6·29-s + 4·31-s + 2·37-s − 6·41-s − 12·47-s − 7·49-s + 2·53-s + 55-s − 4·59-s + 10·61-s − 65-s − 4·67-s − 12·71-s − 2·73-s − 16·79-s − 12·83-s − 2·85-s − 2·89-s + 4·95-s + 10·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s − 0.277·13-s − 0.485·17-s + 0.917·19-s + 1/5·25-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.937·41-s − 1.75·47-s − 49-s + 0.274·53-s + 0.134·55-s − 0.520·59-s + 1.28·61-s − 0.124·65-s − 0.488·67-s − 1.42·71-s − 0.234·73-s − 1.80·79-s − 1.31·83-s − 0.216·85-s − 0.211·89-s + 0.410·95-s + 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−14.02925510175510, −13.43951470223876, −13.04893865736202, −12.60238514487517, −11.88649053359301, −11.53687778777021, −11.23121536491394, −10.27882518444566, −10.04690859341661, −9.715963866185790, −8.905537767648837, −8.623964008152057, −8.040613975736733, −7.383789078920434, −6.915153173365753, −6.366298948015979, −5.934175153963102, −5.252004832952519, −4.661709681213531, −4.373093034228751, −3.287578976440577, −3.094224535716497, −2.263069240386617, −1.597518101145812, −0.9694242766697917, 0,
0.9694242766697917, 1.597518101145812, 2.263069240386617, 3.094224535716497, 3.287578976440577, 4.373093034228751, 4.661709681213531, 5.252004832952519, 5.934175153963102, 6.366298948015979, 6.915153173365753, 7.383789078920434, 8.040613975736733, 8.623964008152057, 8.905537767648837, 9.715963866185790, 10.04690859341661, 10.27882518444566, 11.23121536491394, 11.53687778777021, 11.88649053359301, 12.60238514487517, 13.04893865736202, 13.43951470223876, 14.02925510175510
