| L(s) = 1 | + 5-s − 11-s + 6·13-s + 4·17-s − 6·19-s − 3·23-s − 4·25-s − 4·29-s − 9·31-s − 7·37-s + 2·41-s − 6·43-s − 12·47-s − 7·49-s + 2·53-s − 55-s + 9·59-s − 8·61-s + 6·65-s + 15·67-s + 3·71-s − 6·73-s − 6·79-s − 6·83-s + 4·85-s + 5·89-s − 6·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.301·11-s + 1.66·13-s + 0.970·17-s − 1.37·19-s − 0.625·23-s − 4/5·25-s − 0.742·29-s − 1.61·31-s − 1.15·37-s + 0.312·41-s − 0.914·43-s − 1.75·47-s − 49-s + 0.274·53-s − 0.134·55-s + 1.17·59-s − 1.02·61-s + 0.744·65-s + 1.83·67-s + 0.356·71-s − 0.702·73-s − 0.675·79-s − 0.658·83-s + 0.433·85-s + 0.529·89-s − 0.615·95-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 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−7.79676701841548627476890144597, −6.89740199687044082706390524205, −6.13196927945442026729463587507, −5.71173338309807768933528393739, −4.89830103587644911714431491777, −3.75422324333238215744687908210, −3.47136120482450466989266829443, −2.08378954603136092718150591799, −1.51543774024277585083188088057, 0,
1.51543774024277585083188088057, 2.08378954603136092718150591799, 3.47136120482450466989266829443, 3.75422324333238215744687908210, 4.89830103587644911714431491777, 5.71173338309807768933528393739, 6.13196927945442026729463587507, 6.89740199687044082706390524205, 7.79676701841548627476890144597
