| L(s) = 1 | − 3-s + 2.93·5-s + 7-s + 9-s + 3.68·11-s − 6.61·13-s − 2.93·15-s − 4.18·17-s − 1.17·19-s − 21-s + 23-s + 3.61·25-s − 27-s + 1.68·29-s + 1.17·31-s − 3.68·33-s + 2.93·35-s − 2.31·37-s + 6.61·39-s − 7.17·41-s + 2.74·43-s + 2.93·45-s − 12.2·47-s + 49-s + 4.18·51-s − 13.1·53-s + 10.8·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.31·5-s + 0.377·7-s + 0.333·9-s + 1.10·11-s − 1.83·13-s − 0.757·15-s − 1.01·17-s − 0.269·19-s − 0.218·21-s + 0.208·23-s + 0.723·25-s − 0.192·27-s + 0.312·29-s + 0.210·31-s − 0.640·33-s + 0.496·35-s − 0.381·37-s + 1.05·39-s − 1.12·41-s + 0.418·43-s + 0.437·45-s − 1.78·47-s + 0.142·49-s + 0.586·51-s − 1.81·53-s + 1.45·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2.93T + 5T^{2} \) |
| 11 | \( 1 - 3.68T + 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 17 | \( 1 + 4.18T + 17T^{2} \) |
| 19 | \( 1 + 1.17T + 19T^{2} \) |
| 29 | \( 1 - 1.68T + 29T^{2} \) |
| 31 | \( 1 - 1.17T + 31T^{2} \) |
| 37 | \( 1 + 2.31T + 37T^{2} \) |
| 41 | \( 1 + 7.17T + 41T^{2} \) |
| 43 | \( 1 - 2.74T + 43T^{2} \) |
| 47 | \( 1 + 12.2T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 - 3.06T + 59T^{2} \) |
| 61 | \( 1 + 0.745T + 61T^{2} \) |
| 67 | \( 1 + 8.46T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 6.18T + 79T^{2} \) |
| 83 | \( 1 + 1.33T + 83T^{2} \) |
| 89 | \( 1 - 6.14T + 89T^{2} \) |
| 97 | \( 1 + 6.69T + 97T^{2} \) |
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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
−7.31904438803778583503046812514, −6.55359542014422620599201444227, −6.36353945236985926590843048538, −5.27137477934515055466974454360, −4.90296606312278505527734519319, −4.17031416480320096000324276464, −2.92727112193938768399828031311, −2.04881613349570059573026247163, −1.46188195437792208920578892405, 0,
1.46188195437792208920578892405, 2.04881613349570059573026247163, 2.92727112193938768399828031311, 4.17031416480320096000324276464, 4.90296606312278505527734519319, 5.27137477934515055466974454360, 6.36353945236985926590843048538, 6.55359542014422620599201444227, 7.31904438803778583503046812514
