| L(s) = 1 | + 3·3-s + 12·7-s + 9·9-s + 20·11-s + 58·13-s + 70·17-s + 92·19-s + 36·21-s + 112·23-s + 27·27-s + 66·29-s + 108·31-s + 60·33-s + 58·37-s + 174·39-s + 66·41-s − 388·43-s − 408·47-s − 199·49-s + 210·51-s − 474·53-s + 276·57-s + 540·59-s + 14·61-s + 108·63-s − 276·67-s + 336·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.647·7-s + 1/3·9-s + 0.548·11-s + 1.23·13-s + 0.998·17-s + 1.11·19-s + 0.374·21-s + 1.01·23-s + 0.192·27-s + 0.422·29-s + 0.625·31-s + 0.316·33-s + 0.257·37-s + 0.714·39-s + 0.251·41-s − 1.37·43-s − 1.26·47-s − 0.580·49-s + 0.576·51-s − 1.22·53-s + 0.641·57-s + 1.19·59-s + 0.0293·61-s + 0.215·63-s − 0.503·67-s + 0.586·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.194275284\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.194275284\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 - 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 70 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 112 T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 - 108 T + p^{3} T^{2} \) |
| 37 | \( 1 - 58 T + p^{3} T^{2} \) |
| 41 | \( 1 - 66 T + p^{3} T^{2} \) |
| 43 | \( 1 + 388 T + p^{3} T^{2} \) |
| 47 | \( 1 + 408 T + p^{3} T^{2} \) |
| 53 | \( 1 + 474 T + p^{3} T^{2} \) |
| 59 | \( 1 - 540 T + p^{3} T^{2} \) |
| 61 | \( 1 - 14 T + p^{3} T^{2} \) |
| 67 | \( 1 + 276 T + p^{3} T^{2} \) |
| 71 | \( 1 - 96 T + p^{3} T^{2} \) |
| 73 | \( 1 - 790 T + p^{3} T^{2} \) |
| 79 | \( 1 + 308 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1210 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1426 T + p^{3} T^{2} \) |
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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
−8.378643493191009768426500198309, −8.125331108706174738859895133688, −7.12765339940844409437997374527, −6.38062887526102141737052911684, −5.39501459897258131754353102773, −4.62555374778315238576605785548, −3.56488725154440655029583918005, −3.00721790677572748249896673940, −1.58720507067152332519676249121, −1.00626566377745935887693510915,
1.00626566377745935887693510915, 1.58720507067152332519676249121, 3.00721790677572748249896673940, 3.56488725154440655029583918005, 4.62555374778315238576605785548, 5.39501459897258131754353102773, 6.38062887526102141737052911684, 7.12765339940844409437997374527, 8.125331108706174738859895133688, 8.378643493191009768426500198309
