| L(s) = 1 | − 5-s + 6·11-s + 13-s + 4·17-s − 4·19-s − 4·23-s + 25-s − 4·29-s − 8·31-s − 6·37-s − 10·41-s + 12·43-s + 2·47-s − 7·49-s − 12·53-s − 6·55-s + 2·59-s − 14·61-s − 65-s − 4·67-s + 6·71-s + 10·73-s + 8·79-s − 6·83-s − 4·85-s + 2·89-s + 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.80·11-s + 0.277·13-s + 0.970·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.742·29-s − 1.43·31-s − 0.986·37-s − 1.56·41-s + 1.82·43-s + 0.291·47-s − 49-s − 1.64·53-s − 0.809·55-s + 0.260·59-s − 1.79·61-s − 0.124·65-s − 0.488·67-s + 0.712·71-s + 1.17·73-s + 0.900·79-s − 0.658·83-s − 0.433·85-s + 0.211·89-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 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
−7.41633469552582203402256643432, −6.59753905753911166167850452117, −6.15364124770906443402251538884, −5.33495496533692237283085159750, −4.42459574871585198194433518643, −3.73201779829895235526832367992, −3.37005619717618287193603540906, −1.97486385532543851519595385271, −1.32921780028176395776312360341, 0,
1.32921780028176395776312360341, 1.97486385532543851519595385271, 3.37005619717618287193603540906, 3.73201779829895235526832367992, 4.42459574871585198194433518643, 5.33495496533692237283085159750, 6.15364124770906443402251538884, 6.59753905753911166167850452117, 7.41633469552582203402256643432
