| L(s) = 1 | − 2-s − 2.47·3-s + 4-s + 1.17·5-s + 2.47·6-s + 0.539·7-s − 8-s + 3.13·9-s − 1.17·10-s + 11-s − 2.47·12-s + 3.77·13-s − 0.539·14-s − 2.90·15-s + 16-s − 5.84·17-s − 3.13·18-s − 0.884·19-s + 1.17·20-s − 1.33·21-s − 22-s + 5.41·23-s + 2.47·24-s − 3.62·25-s − 3.77·26-s − 0.339·27-s + 0.539·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.43·3-s + 0.5·4-s + 0.525·5-s + 1.01·6-s + 0.204·7-s − 0.353·8-s + 1.04·9-s − 0.371·10-s + 0.301·11-s − 0.715·12-s + 1.04·13-s − 0.144·14-s − 0.751·15-s + 0.250·16-s − 1.41·17-s − 0.739·18-s − 0.202·19-s + 0.262·20-s − 0.291·21-s − 0.213·22-s + 1.12·23-s + 0.505·24-s − 0.724·25-s − 0.739·26-s − 0.0653·27-s + 0.102·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 - 1.17T + 5T^{2} \) |
| 7 | \( 1 - 0.539T + 7T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 + 5.84T + 17T^{2} \) |
| 19 | \( 1 + 0.884T + 19T^{2} \) |
| 23 | \( 1 - 5.41T + 23T^{2} \) |
| 29 | \( 1 + 1.87T + 29T^{2} \) |
| 31 | \( 1 - 1.14T + 31T^{2} \) |
| 37 | \( 1 - 2.52T + 37T^{2} \) |
| 41 | \( 1 + 1.15T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 - 0.607T + 47T^{2} \) |
| 53 | \( 1 - 2.28T + 53T^{2} \) |
| 59 | \( 1 + 7.45T + 59T^{2} \) |
| 61 | \( 1 + 8.34T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 6.18T + 71T^{2} \) |
| 73 | \( 1 - 12.6T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 - 13.5T + 83T^{2} \) |
| 89 | \( 1 + 4.22T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−8.060546975919410559054013160095, −7.09257682560248279103356407366, −6.31233577604533318957886697887, −6.17998763654703403985991183001, −5.14972158723694171969028564948, −4.49680992254437592564757131555, −3.33504280592832394069404712920, −2.02250583449842028343560963001, −1.19069984244191209177027219927, 0,
1.19069984244191209177027219927, 2.02250583449842028343560963001, 3.33504280592832394069404712920, 4.49680992254437592564757131555, 5.14972158723694171969028564948, 6.17998763654703403985991183001, 6.31233577604533318957886697887, 7.09257682560248279103356407366, 8.060546975919410559054013160095
