| L(s) = 1 | + 5-s + 2·7-s − 4·11-s − 6·13-s − 2·17-s − 8·19-s − 6·23-s + 25-s + 2·29-s − 4·31-s + 2·35-s + 2·37-s + 10·41-s + 2·43-s − 2·47-s − 3·49-s − 2·53-s − 4·55-s + 2·61-s − 6·65-s + 6·67-s − 12·71-s + 10·73-s − 8·77-s + 8·79-s − 10·83-s − 2·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 1.20·11-s − 1.66·13-s − 0.485·17-s − 1.83·19-s − 1.25·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s + 0.338·35-s + 0.328·37-s + 1.56·41-s + 0.304·43-s − 0.291·47-s − 3/7·49-s − 0.274·53-s − 0.539·55-s + 0.256·61-s − 0.744·65-s + 0.733·67-s − 1.42·71-s + 1.17·73-s − 0.911·77-s + 0.900·79-s − 1.09·83-s − 0.216·85-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 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−9.173036644566947320848099598200, −8.165910020926354745044007473450, −7.69912747069076488354294405346, −6.68490910721201518672569982184, −5.73515780171722867164156586914, −4.89165840272277024202803368024, −4.22669245032069068079121236742, −2.55409854567901025018681303094, −2.04771450425397713697384155116, 0,
2.04771450425397713697384155116, 2.55409854567901025018681303094, 4.22669245032069068079121236742, 4.89165840272277024202803368024, 5.73515780171722867164156586914, 6.68490910721201518672569982184, 7.69912747069076488354294405346, 8.165910020926354745044007473450, 9.173036644566947320848099598200
