L(s) = 1 | + 2·5-s + 11-s − 2·13-s − 6·17-s − 4·23-s − 25-s + 2·29-s + 10·37-s − 6·41-s + 8·43-s + 4·47-s − 7·49-s − 6·53-s + 2·55-s − 12·59-s − 2·61-s − 4·65-s − 4·67-s − 12·71-s − 14·73-s + 16·79-s − 12·83-s − 12·85-s − 10·89-s − 14·97-s − 6·101-s + 8·103-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.301·11-s − 0.554·13-s − 1.45·17-s − 0.834·23-s − 1/5·25-s + 0.371·29-s + 1.64·37-s − 0.937·41-s + 1.21·43-s + 0.583·47-s − 49-s − 0.824·53-s + 0.269·55-s − 1.56·59-s − 0.256·61-s − 0.496·65-s − 0.488·67-s − 1.42·71-s − 1.63·73-s + 1.80·79-s − 1.31·83-s − 1.30·85-s − 1.05·89-s − 1.42·97-s − 0.597·101-s + 0.788·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.66801225265299385870670777834, −6.89723242969747737031934892174, −6.15379190336573720717398861429, −5.77488688886590375137826764683, −4.64052519860568286949163704744, −4.26984581901828839645286801938, −3.01510013803481993013087415181, −2.26606747679282501943052738688, −1.49003574627855910269883202884, 0,
1.49003574627855910269883202884, 2.26606747679282501943052738688, 3.01510013803481993013087415181, 4.26984581901828839645286801938, 4.64052519860568286949163704744, 5.77488688886590375137826764683, 6.15379190336573720717398861429, 6.89723242969747737031934892174, 7.66801225265299385870670777834