| L(s) = 1 | − 3·3-s − 5·5-s + 10·7-s + 9·9-s + 22·11-s − 26·13-s + 15·15-s + 14·17-s + 34·19-s − 30·21-s − 190·23-s + 25·25-s − 27·27-s + 162·29-s − 268·31-s − 66·33-s − 50·35-s + 362·37-s + 78·39-s − 170·41-s − 16·43-s − 45·45-s + 434·47-s − 243·49-s − 42·51-s − 594·53-s − 110·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.539·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 0.258·15-s + 0.199·17-s + 0.410·19-s − 0.311·21-s − 1.72·23-s + 1/5·25-s − 0.192·27-s + 1.03·29-s − 1.55·31-s − 0.348·33-s − 0.241·35-s + 1.60·37-s + 0.320·39-s − 0.647·41-s − 0.0567·43-s − 0.149·45-s + 1.34·47-s − 0.708·49-s − 0.115·51-s − 1.53·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 34 T + p^{3} T^{2} \) |
| 23 | \( 1 + 190 T + p^{3} T^{2} \) |
| 29 | \( 1 - 162 T + p^{3} T^{2} \) |
| 31 | \( 1 + 268 T + p^{3} T^{2} \) |
| 37 | \( 1 - 362 T + p^{3} T^{2} \) |
| 41 | \( 1 + 170 T + p^{3} T^{2} \) |
| 43 | \( 1 + 16 T + p^{3} T^{2} \) |
| 47 | \( 1 - 434 T + p^{3} T^{2} \) |
| 53 | \( 1 + 594 T + p^{3} T^{2} \) |
| 59 | \( 1 - 170 T + p^{3} T^{2} \) |
| 61 | \( 1 + 130 T + p^{3} T^{2} \) |
| 67 | \( 1 - 1024 T + p^{3} T^{2} \) |
| 71 | \( 1 - 280 T + p^{3} T^{2} \) |
| 73 | \( 1 - 282 T + p^{3} T^{2} \) |
| 79 | \( 1 + 160 T + p^{3} T^{2} \) |
| 83 | \( 1 - 732 T + p^{3} T^{2} \) |
| 89 | \( 1 + 746 T + p^{3} T^{2} \) |
| 97 | \( 1 + 534 T + p^{3} T^{2} \) |
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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
−8.250967153808124226188306017960, −7.73468687643735773657214219371, −6.86299961364120779400123843163, −6.05091232904216984515813895098, −5.19018720213719029486413291870, −4.37769624979295814555096387102, −3.60519364123077157037064578652, −2.26462234722605672610817593788, −1.16787284993251330959563082653, 0,
1.16787284993251330959563082653, 2.26462234722605672610817593788, 3.60519364123077157037064578652, 4.37769624979295814555096387102, 5.19018720213719029486413291870, 6.05091232904216984515813895098, 6.86299961364120779400123843163, 7.73468687643735773657214219371, 8.250967153808124226188306017960
