| L(s) = 1 | + 12·3-s + 25·5-s + 49·7-s − 99·9-s − 556·11-s − 354·13-s + 300·15-s + 770·17-s + 2.68e3·19-s + 588·21-s + 1.52e3·23-s + 625·25-s − 4.10e3·27-s − 2.41e3·29-s − 7.84e3·31-s − 6.67e3·33-s + 1.22e3·35-s − 314·37-s − 4.24e3·39-s − 1.78e4·41-s − 1.64e4·43-s − 2.47e3·45-s − 5.37e3·47-s + 2.40e3·49-s + 9.24e3·51-s + 1.65e3·53-s − 1.39e4·55-s + ⋯ |
| L(s) = 1 | + 0.769·3-s + 0.447·5-s + 0.377·7-s − 0.407·9-s − 1.38·11-s − 0.580·13-s + 0.344·15-s + 0.646·17-s + 1.70·19-s + 0.290·21-s + 0.602·23-s + 1/5·25-s − 1.08·27-s − 0.533·29-s − 1.46·31-s − 1.06·33-s + 0.169·35-s − 0.0377·37-s − 0.447·39-s − 1.66·41-s − 1.35·43-s − 0.182·45-s − 0.354·47-s + 1/7·49-s + 0.497·51-s + 0.0808·53-s − 0.619·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 4 p T + p^{5} T^{2} \) |
| 11 | \( 1 + 556 T + p^{5} T^{2} \) |
| 13 | \( 1 + 354 T + p^{5} T^{2} \) |
| 17 | \( 1 - 770 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2684 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1528 T + p^{5} T^{2} \) |
| 29 | \( 1 + 2418 T + p^{5} T^{2} \) |
| 31 | \( 1 + 7840 T + p^{5} T^{2} \) |
| 37 | \( 1 + 314 T + p^{5} T^{2} \) |
| 41 | \( 1 + 17878 T + p^{5} T^{2} \) |
| 43 | \( 1 + 16476 T + p^{5} T^{2} \) |
| 47 | \( 1 + 5376 T + p^{5} T^{2} \) |
| 53 | \( 1 - 1654 T + p^{5} T^{2} \) |
| 59 | \( 1 - 29492 T + p^{5} T^{2} \) |
| 61 | \( 1 - 27630 T + p^{5} T^{2} \) |
| 67 | \( 1 + 57716 T + p^{5} T^{2} \) |
| 71 | \( 1 + 70648 T + p^{5} T^{2} \) |
| 73 | \( 1 - 74202 T + p^{5} T^{2} \) |
| 79 | \( 1 + 74336 T + p^{5} T^{2} \) |
| 83 | \( 1 + 44068 T + p^{5} T^{2} \) |
| 89 | \( 1 - 129306 T + p^{5} T^{2} \) |
| 97 | \( 1 + 137646 T + p^{5} T^{2} \) |
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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
−9.557530521745608761694103826387, −8.616715155763640354674972196873, −7.79918915822145031923993201504, −7.13151972899997533282759773903, −5.46226088113655834022048188567, −5.19337761059366561818304077485, −3.44149971238919347364439242022, −2.70850303181764672086126711297, −1.60491187551803143670716240568, 0,
1.60491187551803143670716240568, 2.70850303181764672086126711297, 3.44149971238919347364439242022, 5.19337761059366561818304077485, 5.46226088113655834022048188567, 7.13151972899997533282759773903, 7.79918915822145031923993201504, 8.616715155763640354674972196873, 9.557530521745608761694103826387
