| L(s) = 1 | + 2-s + 4-s + 5-s − 2·7-s + 8-s + 10-s − 3·11-s + 13-s − 2·14-s + 16-s − 3·19-s + 20-s − 3·22-s − 23-s + 25-s + 26-s − 2·28-s − 29-s − 31-s + 32-s − 2·35-s − 5·37-s − 3·38-s + 40-s − 2·41-s − 9·43-s − 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.755·7-s + 0.353·8-s + 0.316·10-s − 0.904·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.688·19-s + 0.223·20-s − 0.639·22-s − 0.208·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 0.185·29-s − 0.179·31-s + 0.176·32-s − 0.338·35-s − 0.821·37-s − 0.486·38-s + 0.158·40-s − 0.312·41-s − 1.37·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−13.11398778411688, −12.87835812067525, −12.11440705185717, −11.81353322852151, −11.28380094211271, −10.68478014526132, −10.27539967373066, −10.12015346402768, −9.395479638520480, −8.972456861459307, −8.418613079234716, −7.922263464172849, −7.451632042201429, −6.825889970600480, −6.426300136794862, −6.081184821862306, −5.543444825908602, −4.956202512602099, −4.706520001096236, −3.907522535760111, −3.456191250036623, −2.910416380574977, −2.559362449274663, −1.660091511881551, −1.473358403678311, 0, 0,
1.473358403678311, 1.660091511881551, 2.559362449274663, 2.910416380574977, 3.456191250036623, 3.907522535760111, 4.706520001096236, 4.956202512602099, 5.543444825908602, 6.081184821862306, 6.426300136794862, 6.825889970600480, 7.451632042201429, 7.922263464172849, 8.418613079234716, 8.972456861459307, 9.395479638520480, 10.12015346402768, 10.27539967373066, 10.68478014526132, 11.28380094211271, 11.81353322852151, 12.11440705185717, 12.87835812067525, 13.11398778411688
