| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s − 3·11-s + 6·13-s − 15-s + 17-s − 6·19-s − 21-s + 4·23-s + 25-s − 27-s − 5·29-s − 9·31-s + 3·33-s + 35-s + 37-s − 6·39-s − 3·41-s + 7·43-s + 45-s − 6·49-s − 51-s − 53-s − 3·55-s + 6·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s − 0.904·11-s + 1.66·13-s − 0.258·15-s + 0.242·17-s − 1.37·19-s − 0.218·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.928·29-s − 1.61·31-s + 0.522·33-s + 0.169·35-s + 0.164·37-s − 0.960·39-s − 0.468·41-s + 1.06·43-s + 0.149·45-s − 6/7·49-s − 0.140·51-s − 0.137·53-s − 0.404·55-s + 0.794·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−7.40113423955929323548007832338, −6.60109550139874865903647562944, −5.90541387576366586423628576468, −5.51751466992329370196556091060, −4.68922036529989543100465034148, −3.93408626620530963961367984250, −3.09138553723886275461557057129, −2.00705671822820973043717060586, −1.30440359817692274183100841923, 0,
1.30440359817692274183100841923, 2.00705671822820973043717060586, 3.09138553723886275461557057129, 3.93408626620530963961367984250, 4.68922036529989543100465034148, 5.51751466992329370196556091060, 5.90541387576366586423628576468, 6.60109550139874865903647562944, 7.40113423955929323548007832338
