| L(s) = 1 | + 1.41·3-s + 5-s + 2.44·7-s − 0.999·9-s − 1.41·11-s − 5.46·13-s + 1.41·15-s + 0.732·17-s − 2.44·19-s + 3.46·21-s − 6.31·23-s + 25-s − 5.65·27-s − 29-s − 5.27·31-s − 2.00·33-s + 2.44·35-s + 7.66·37-s − 7.72·39-s − 9.46·41-s − 3.48·43-s − 0.999·45-s − 9.14·47-s − 1.00·49-s + 1.03·51-s + 5.46·53-s − 1.41·55-s + ⋯ |
| L(s) = 1 | + 0.816·3-s + 0.447·5-s + 0.925·7-s − 0.333·9-s − 0.426·11-s − 1.51·13-s + 0.365·15-s + 0.177·17-s − 0.561·19-s + 0.755·21-s − 1.31·23-s + 0.200·25-s − 1.08·27-s − 0.185·29-s − 0.947·31-s − 0.348·33-s + 0.414·35-s + 1.25·37-s − 1.23·39-s − 1.47·41-s − 0.531·43-s − 0.149·45-s − 1.33·47-s − 0.142·49-s + 0.144·51-s + 0.750·53-s − 0.190·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 - 2.44T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 - 0.732T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 + 6.31T + 23T^{2} \) |
| 31 | \( 1 + 5.27T + 31T^{2} \) |
| 37 | \( 1 - 7.66T + 37T^{2} \) |
| 41 | \( 1 + 9.46T + 41T^{2} \) |
| 43 | \( 1 + 3.48T + 43T^{2} \) |
| 47 | \( 1 + 9.14T + 47T^{2} \) |
| 53 | \( 1 - 5.46T + 53T^{2} \) |
| 59 | \( 1 - 8.76T + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 3.48T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 - 0.656T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 0.928T + 89T^{2} \) |
| 97 | \( 1 - 15.1T + 97T^{2} \) |
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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
−8.022976016734778707590517228301, −7.48919797088208210078686220938, −6.56423652952318136852388982535, −5.58804078287359237675853671123, −5.03957674117244925082029894186, −4.19577015547146930444025212853, −3.18140653713423558931918401319, −2.28798554570515309578857698590, −1.81620609391331703388951221371, 0,
1.81620609391331703388951221371, 2.28798554570515309578857698590, 3.18140653713423558931918401319, 4.19577015547146930444025212853, 5.03957674117244925082029894186, 5.58804078287359237675853671123, 6.56423652952318136852388982535, 7.48919797088208210078686220938, 8.022976016734778707590517228301
