| L(s) = 1 | + 3-s − 2·5-s − 7-s − 2·9-s − 11-s − 6·13-s − 2·15-s − 4·17-s + 8·19-s − 21-s − 6·23-s − 25-s − 5·27-s + 2·29-s + 4·31-s − 33-s + 2·35-s − 37-s − 6·39-s + 7·41-s − 2·43-s + 4·45-s − 9·47-s − 6·49-s − 4·51-s − 3·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 1.66·13-s − 0.516·15-s − 0.970·17-s + 1.83·19-s − 0.218·21-s − 1.25·23-s − 1/5·25-s − 0.962·27-s + 0.371·29-s + 0.718·31-s − 0.174·33-s + 0.338·35-s − 0.164·37-s − 0.960·39-s + 1.09·41-s − 0.304·43-s + 0.596·45-s − 1.31·47-s − 6/7·49-s − 0.560·51-s − 0.412·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−9.969562582054726543474486278240, −9.494663490575910035621177927785, −8.289160116966715982503627200518, −7.75456110220713984165022091972, −6.86682095955105512061953544939, −5.55163436815551852246094638061, −4.49161097993379934705251491217, −3.32391285921883791433244095783, −2.42130817560810783959503841896, 0,
2.42130817560810783959503841896, 3.32391285921883791433244095783, 4.49161097993379934705251491217, 5.55163436815551852246094638061, 6.86682095955105512061953544939, 7.75456110220713984165022091972, 8.289160116966715982503627200518, 9.494663490575910035621177927785, 9.969562582054726543474486278240
