| L(s) = 1 | − 3.67·2-s + 5.54·4-s + 5·5-s + 9.04·8-s − 18.3·10-s − 65.0·11-s + 6.87·13-s − 77.6·16-s + 68.0·17-s − 23.5·19-s + 27.7·20-s + 239.·22-s + 5.49·23-s + 25·25-s − 25.2·26-s + 138.·29-s − 251.·31-s + 213.·32-s − 250.·34-s + 127.·37-s + 86.6·38-s + 45.2·40-s − 126.·41-s + 91.5·43-s − 360.·44-s − 20.2·46-s + 568.·47-s + ⋯ |
| L(s) = 1 | − 1.30·2-s + 0.692·4-s + 0.447·5-s + 0.399·8-s − 0.581·10-s − 1.78·11-s + 0.146·13-s − 1.21·16-s + 0.970·17-s − 0.284·19-s + 0.309·20-s + 2.32·22-s + 0.0498·23-s + 0.200·25-s − 0.190·26-s + 0.887·29-s − 1.45·31-s + 1.17·32-s − 1.26·34-s + 0.566·37-s + 0.369·38-s + 0.178·40-s − 0.482·41-s + 0.324·43-s − 1.23·44-s − 0.0648·46-s + 1.76·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 3.67T + 8T^{2} \) |
| 11 | \( 1 + 65.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 6.87T + 2.19e3T^{2} \) |
| 17 | \( 1 - 68.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 23.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 5.49T + 1.21e4T^{2} \) |
| 29 | \( 1 - 138.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 127.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 126.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 91.5T + 7.95e4T^{2} \) |
| 47 | \( 1 - 568.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 590.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 509.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 261.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 922.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 519.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.06e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 112.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 593.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 242.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.53e3T + 9.12e5T^{2} \) |
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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
−8.504846150690218260271362183662, −7.53533850923646512325649332175, −7.28752528880331174790580824179, −5.94242621541965389159745897204, −5.31378382962305874662004112531, −4.31489057442169416669584009971, −2.94779374866121425269536777391, −2.11546787319039043900733033894, −1.01647524392983972251650366331, 0,
1.01647524392983972251650366331, 2.11546787319039043900733033894, 2.94779374866121425269536777391, 4.31489057442169416669584009971, 5.31378382962305874662004112531, 5.94242621541965389159745897204, 7.28752528880331174790580824179, 7.53533850923646512325649332175, 8.504846150690218260271362183662
