| L(s) = 1 | − 8·3-s + 5·5-s − 24·7-s + 37·9-s + 11·11-s − 22·13-s − 40·15-s + 22·17-s + 28·19-s + 192·21-s + 44·23-s + 25·25-s − 80·27-s + 110·29-s + 40·31-s − 88·33-s − 120·35-s − 362·37-s + 176·39-s + 210·41-s − 260·43-s + 185·45-s + 460·47-s + 233·49-s − 176·51-s + 662·53-s + 55·55-s + ⋯ |
| L(s) = 1 | − 1.53·3-s + 0.447·5-s − 1.29·7-s + 1.37·9-s + 0.301·11-s − 0.469·13-s − 0.688·15-s + 0.313·17-s + 0.338·19-s + 1.99·21-s + 0.398·23-s + 1/5·25-s − 0.570·27-s + 0.704·29-s + 0.231·31-s − 0.464·33-s − 0.579·35-s − 1.60·37-s + 0.722·39-s + 0.799·41-s − 0.922·43-s + 0.612·45-s + 1.42·47-s + 0.679·49-s − 0.483·51-s + 1.71·53-s + 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 11 | \( 1 - p T \) |
| good | 3 | \( 1 + 8 T + p^{3} T^{2} \) |
| 7 | \( 1 + 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 22 T + p^{3} T^{2} \) |
| 19 | \( 1 - 28 T + p^{3} T^{2} \) |
| 23 | \( 1 - 44 T + p^{3} T^{2} \) |
| 29 | \( 1 - 110 T + p^{3} T^{2} \) |
| 31 | \( 1 - 40 T + p^{3} T^{2} \) |
| 37 | \( 1 + 362 T + p^{3} T^{2} \) |
| 41 | \( 1 - 210 T + p^{3} T^{2} \) |
| 43 | \( 1 + 260 T + p^{3} T^{2} \) |
| 47 | \( 1 - 460 T + p^{3} T^{2} \) |
| 53 | \( 1 - 662 T + p^{3} T^{2} \) |
| 59 | \( 1 - 68 T + p^{3} T^{2} \) |
| 61 | \( 1 - 606 T + p^{3} T^{2} \) |
| 67 | \( 1 - 312 T + p^{3} T^{2} \) |
| 71 | \( 1 + 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1042 T + p^{3} T^{2} \) |
| 79 | \( 1 - 552 T + p^{3} T^{2} \) |
| 83 | \( 1 + 268 T + p^{3} T^{2} \) |
| 89 | \( 1 + 966 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1334 T + p^{3} T^{2} \) |
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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
−9.672261375740525937519564161529, −8.657393350827599009639327674907, −7.10395304486891778039961373728, −6.70535236115837156423619005317, −5.77812953804370582908707581143, −5.21808075066702718098366322244, −4.01165159809247119859856265816, −2.75181353423915078599752114495, −1.09685180899579126968711831767, 0,
1.09685180899579126968711831767, 2.75181353423915078599752114495, 4.01165159809247119859856265816, 5.21808075066702718098366322244, 5.77812953804370582908707581143, 6.70535236115837156423619005317, 7.10395304486891778039961373728, 8.657393350827599009639327674907, 9.672261375740525937519564161529
