| L(s) = 1 | − 3·3-s + 23·7-s + 9·9-s + 30·11-s − 29·13-s − 78·17-s − 149·19-s − 69·21-s + 150·23-s − 27·27-s − 234·29-s + 217·31-s − 90·33-s − 146·37-s + 87·39-s − 156·41-s − 433·43-s + 30·47-s + 186·49-s + 234·51-s + 552·53-s + 447·57-s + 270·59-s + 275·61-s + 207·63-s + 803·67-s − 450·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.24·7-s + 1/3·9-s + 0.822·11-s − 0.618·13-s − 1.11·17-s − 1.79·19-s − 0.717·21-s + 1.35·23-s − 0.192·27-s − 1.49·29-s + 1.25·31-s − 0.474·33-s − 0.648·37-s + 0.357·39-s − 0.594·41-s − 1.53·43-s + 0.0931·47-s + 0.542·49-s + 0.642·51-s + 1.43·53-s + 1.03·57-s + 0.595·59-s + 0.577·61-s + 0.413·63-s + 1.46·67-s − 0.785·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 23 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 29 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 149 T + p^{3} T^{2} \) |
| 23 | \( 1 - 150 T + p^{3} T^{2} \) |
| 29 | \( 1 + 234 T + p^{3} T^{2} \) |
| 31 | \( 1 - 7 p T + p^{3} T^{2} \) |
| 37 | \( 1 + 146 T + p^{3} T^{2} \) |
| 41 | \( 1 + 156 T + p^{3} T^{2} \) |
| 43 | \( 1 + 433 T + p^{3} T^{2} \) |
| 47 | \( 1 - 30 T + p^{3} T^{2} \) |
| 53 | \( 1 - 552 T + p^{3} T^{2} \) |
| 59 | \( 1 - 270 T + p^{3} T^{2} \) |
| 61 | \( 1 - 275 T + p^{3} T^{2} \) |
| 67 | \( 1 - 803 T + p^{3} T^{2} \) |
| 71 | \( 1 + 660 T + p^{3} T^{2} \) |
| 73 | \( 1 - 646 T + p^{3} T^{2} \) |
| 79 | \( 1 + 992 T + p^{3} T^{2} \) |
| 83 | \( 1 + 846 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1488 T + p^{3} T^{2} \) |
| 97 | \( 1 - 319 T + p^{3} T^{2} \) |
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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
−8.778145697277545656962082277758, −8.340788227157342300129272297821, −7.06164258134474406633670665312, −6.63825711729998090155366861819, −5.42919498797208678486596067616, −4.68355835472474856160439368282, −3.97858183440676667674887731493, −2.33534985404933562110557273908, −1.41911325686009124144875530521, 0,
1.41911325686009124144875530521, 2.33534985404933562110557273908, 3.97858183440676667674887731493, 4.68355835472474856160439368282, 5.42919498797208678486596067616, 6.63825711729998090155366861819, 7.06164258134474406633670665312, 8.340788227157342300129272297821, 8.778145697277545656962082277758
