| L(s) = 1 | − 5-s − 2·7-s + 2·11-s + 2·17-s − 4·19-s + 25-s − 2·29-s − 8·31-s + 2·35-s − 4·37-s + 8·41-s − 8·43-s − 8·47-s − 3·49-s − 10·53-s − 2·55-s − 6·59-s + 2·61-s − 12·67-s + 12·71-s − 2·73-s − 4·77-s − 8·79-s − 4·83-s − 2·85-s + 12·89-s + 4·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.603·11-s + 0.485·17-s − 0.917·19-s + 1/5·25-s − 0.371·29-s − 1.43·31-s + 0.338·35-s − 0.657·37-s + 1.24·41-s − 1.21·43-s − 1.16·47-s − 3/7·49-s − 1.37·53-s − 0.269·55-s − 0.781·59-s + 0.256·61-s − 1.46·67-s + 1.42·71-s − 0.234·73-s − 0.455·77-s − 0.900·79-s − 0.439·83-s − 0.216·85-s + 1.27·89-s + 0.410·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.192602683771207107231383355011, −8.353789705562506406524679643643, −7.48616061685451849552484937991, −6.66547726825929901514782306763, −5.96534369241922370325330817971, −4.85681933287367456280527642087, −3.84388617445388444670245263060, −3.13315432851376326757177285673, −1.70160744604485133150386942637, 0,
1.70160744604485133150386942637, 3.13315432851376326757177285673, 3.84388617445388444670245263060, 4.85681933287367456280527642087, 5.96534369241922370325330817971, 6.66547726825929901514782306763, 7.48616061685451849552484937991, 8.353789705562506406524679643643, 9.192602683771207107231383355011
