| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s − 2·11-s + 15-s − 4·21-s − 8·23-s + 25-s + 27-s + 2·29-s − 2·31-s − 2·33-s − 4·35-s − 8·37-s − 2·41-s + 4·43-s + 45-s + 9·49-s − 6·53-s − 2·55-s − 14·59-s − 14·61-s − 4·63-s − 4·67-s − 8·69-s + 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.258·15-s − 0.872·21-s − 1.66·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.359·31-s − 0.348·33-s − 0.676·35-s − 1.31·37-s − 0.312·41-s + 0.609·43-s + 0.149·45-s + 9/7·49-s − 0.824·53-s − 0.269·55-s − 1.82·59-s − 1.79·61-s − 0.503·63-s − 0.488·67-s − 0.963·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.975784482874164468937827444067, −8.041289019730938230132967860356, −7.26926784547186801970456113424, −6.35748243975676630013335276192, −5.81966760462348143512239602629, −4.64099478965301858716726482454, −3.55892389127543084754638209246, −2.88655292546004445931032442741, −1.84416625898130938268214518612, 0,
1.84416625898130938268214518612, 2.88655292546004445931032442741, 3.55892389127543084754638209246, 4.64099478965301858716726482454, 5.81966760462348143512239602629, 6.35748243975676630013335276192, 7.26926784547186801970456113424, 8.041289019730938230132967860356, 8.975784482874164468937827444067
