| L(s) = 1 | + 1.04·2-s − 0.911·4-s + 1.64·5-s + 5.01·7-s − 3.03·8-s + 1.71·10-s + 11-s − 3.01·13-s + 5.22·14-s − 1.34·16-s − 3.18·17-s − 6.03·19-s − 1.50·20-s + 1.04·22-s − 4.89·23-s − 2.28·25-s − 3.14·26-s − 4.56·28-s − 0.904·29-s + 5.23·31-s + 4.67·32-s − 3.32·34-s + 8.26·35-s − 5.38·37-s − 6.29·38-s − 5.00·40-s − 11.7·41-s + ⋯ |
| L(s) = 1 | + 0.737·2-s − 0.455·4-s + 0.737·5-s + 1.89·7-s − 1.07·8-s + 0.543·10-s + 0.301·11-s − 0.837·13-s + 1.39·14-s − 0.336·16-s − 0.773·17-s − 1.38·19-s − 0.336·20-s + 0.222·22-s − 1.02·23-s − 0.456·25-s − 0.617·26-s − 0.863·28-s − 0.167·29-s + 0.940·31-s + 0.825·32-s − 0.570·34-s + 1.39·35-s − 0.884·37-s − 1.02·38-s − 0.791·40-s − 1.82·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.04T + 2T^{2} \) |
| 5 | \( 1 - 1.64T + 5T^{2} \) |
| 7 | \( 1 - 5.01T + 7T^{2} \) |
| 13 | \( 1 + 3.01T + 13T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 + 6.03T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 + 0.904T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + 5.38T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 + 7.10T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 0.468T + 59T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 - 9.71T + 71T^{2} \) |
| 73 | \( 1 + 3.32T + 73T^{2} \) |
| 79 | \( 1 + 5.59T + 79T^{2} \) |
| 83 | \( 1 + 0.444T + 83T^{2} \) |
| 89 | \( 1 - 3.27T + 89T^{2} \) |
| 97 | \( 1 - 7.38T + 97T^{2} \) |
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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
−7.932508177040532782064760011806, −6.77285468478726104681788346198, −6.18613823800681338964887379290, −5.32262475458147993378652655930, −4.73939800242216646090628524685, −4.41949219298007061762914795672, −3.38738288281446843137469652285, −2.10088371631999868954856854062, −1.75940247592014878765046598176, 0,
1.75940247592014878765046598176, 2.10088371631999868954856854062, 3.38738288281446843137469652285, 4.41949219298007061762914795672, 4.73939800242216646090628524685, 5.32262475458147993378652655930, 6.18613823800681338964887379290, 6.77285468478726104681788346198, 7.932508177040532782064760011806
