| L(s) = 1 | + 3-s + 7-s + 9-s + 11-s − 2·13-s − 4·17-s − 6·19-s + 21-s + 4·23-s − 5·25-s + 27-s − 10·29-s − 6·31-s + 33-s − 6·37-s − 2·39-s − 12·41-s + 8·43-s − 2·47-s + 49-s − 4·51-s + 6·53-s − 6·57-s + 8·59-s + 6·61-s + 63-s + 4·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.970·17-s − 1.37·19-s + 0.218·21-s + 0.834·23-s − 25-s + 0.192·27-s − 1.85·29-s − 1.07·31-s + 0.174·33-s − 0.986·37-s − 0.320·39-s − 1.87·41-s + 1.21·43-s − 0.291·47-s + 1/7·49-s − 0.560·51-s + 0.824·53-s − 0.794·57-s + 1.04·59-s + 0.768·61-s + 0.125·63-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−8.336678566584196069711461502340, −7.27799845398591508441996698156, −6.97536064814481970593676772354, −5.88594420769943906193661157841, −5.07834541625290325111035006299, −4.17548335786158687514507027199, −3.57130049186751159493414790655, −2.30654659961582856953896636946, −1.76609846180802589057694463941, 0,
1.76609846180802589057694463941, 2.30654659961582856953896636946, 3.57130049186751159493414790655, 4.17548335786158687514507027199, 5.07834541625290325111035006299, 5.88594420769943906193661157841, 6.97536064814481970593676772354, 7.27799845398591508441996698156, 8.336678566584196069711461502340
