| L(s) = 1 | + 1.84·3-s + 3.17·5-s − 7-s + 0.392·9-s − 4.90·11-s − 5.51·13-s + 5.84·15-s − 17-s − 2.99·19-s − 1.84·21-s − 4.99·23-s + 5.05·25-s − 4.80·27-s − 8.89·29-s + 3.56·31-s − 9.02·33-s − 3.17·35-s + 5.95·37-s − 10.1·39-s + 9.28·41-s + 1.21·43-s + 1.24·45-s − 11.8·47-s + 49-s − 1.84·51-s − 5.42·53-s − 15.5·55-s + ⋯ |
| L(s) = 1 | + 1.06·3-s + 1.41·5-s − 0.377·7-s + 0.130·9-s − 1.47·11-s − 1.53·13-s + 1.50·15-s − 0.242·17-s − 0.687·19-s − 0.401·21-s − 1.04·23-s + 1.01·25-s − 0.924·27-s − 1.65·29-s + 0.640·31-s − 1.57·33-s − 0.536·35-s + 0.978·37-s − 1.62·39-s + 1.45·41-s + 0.185·43-s + 0.185·45-s − 1.72·47-s + 0.142·49-s − 0.257·51-s − 0.745·53-s − 2.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.84T + 3T^{2} \) |
| 5 | \( 1 - 3.17T + 5T^{2} \) |
| 11 | \( 1 + 4.90T + 11T^{2} \) |
| 13 | \( 1 + 5.51T + 13T^{2} \) |
| 19 | \( 1 + 2.99T + 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 + 8.89T + 29T^{2} \) |
| 31 | \( 1 - 3.56T + 31T^{2} \) |
| 37 | \( 1 - 5.95T + 37T^{2} \) |
| 41 | \( 1 - 9.28T + 41T^{2} \) |
| 43 | \( 1 - 1.21T + 43T^{2} \) |
| 47 | \( 1 + 11.8T + 47T^{2} \) |
| 53 | \( 1 + 5.42T + 53T^{2} \) |
| 59 | \( 1 - 0.469T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 14.6T + 67T^{2} \) |
| 71 | \( 1 - 11.0T + 71T^{2} \) |
| 73 | \( 1 + 2.08T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 - 2.17T + 83T^{2} \) |
| 89 | \( 1 + 6.04T + 89T^{2} \) |
| 97 | \( 1 + 7.90T + 97T^{2} \) |
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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
−7.942642523946257844244354649552, −7.74386870118552835099921120709, −6.62480517356833891292518422578, −5.82829985552514166425742467800, −5.23953389235861282530885545470, −4.28500160812462322256130953146, −3.07236572292025164156252737948, −2.34856075262559705032357499827, −2.06010579915212196263334248047, 0,
2.06010579915212196263334248047, 2.34856075262559705032357499827, 3.07236572292025164156252737948, 4.28500160812462322256130953146, 5.23953389235861282530885545470, 5.82829985552514166425742467800, 6.62480517356833891292518422578, 7.74386870118552835099921120709, 7.942642523946257844244354649552
