| L(s) = 1 | + 2-s + 1.23·3-s + 4-s − 1.34·5-s + 1.23·6-s − 1.90·7-s + 8-s − 1.47·9-s − 1.34·10-s − 11-s + 1.23·12-s + 2.92·13-s − 1.90·14-s − 1.65·15-s + 16-s + 2.56·17-s − 1.47·18-s + 1.17·19-s − 1.34·20-s − 2.35·21-s − 22-s − 8.64·23-s + 1.23·24-s − 3.19·25-s + 2.92·26-s − 5.52·27-s − 1.90·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.712·3-s + 0.5·4-s − 0.601·5-s + 0.504·6-s − 0.721·7-s + 0.353·8-s − 0.491·9-s − 0.425·10-s − 0.301·11-s + 0.356·12-s + 0.811·13-s − 0.510·14-s − 0.428·15-s + 0.250·16-s + 0.623·17-s − 0.347·18-s + 0.269·19-s − 0.300·20-s − 0.514·21-s − 0.213·22-s − 1.80·23-s + 0.252·24-s − 0.638·25-s + 0.573·26-s − 1.06·27-s − 0.360·28-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 | \( 1 - 1.23T + 3T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 7 | \( 1 + 1.90T + 7T^{2} \) |
| 13 | \( 1 - 2.92T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + 8.64T + 23T^{2} \) |
| 29 | \( 1 - 6.10T + 29T^{2} \) |
| 31 | \( 1 + 1.59T + 31T^{2} \) |
| 37 | \( 1 + 7.80T + 37T^{2} \) |
| 41 | \( 1 + 7.95T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 + 3.21T + 47T^{2} \) |
| 53 | \( 1 + 0.670T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 0.806T + 61T^{2} \) |
| 67 | \( 1 + 13.6T + 67T^{2} \) |
| 71 | \( 1 - 7.82T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 + 6.11T + 89T^{2} \) |
| 97 | \( 1 - 5.36T + 97T^{2} \) |
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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
−8.046616878275771057379706890968, −7.38002146760642370323109342650, −6.36338767553654384138334354417, −5.89337862977893958941121052384, −4.98436396657798869813125444940, −3.88055635462899308858882741730, −3.49243513695378970249921456071, −2.76985913843848130914349480461, −1.70124834666586452793925005589, 0,
1.70124834666586452793925005589, 2.76985913843848130914349480461, 3.49243513695378970249921456071, 3.88055635462899308858882741730, 4.98436396657798869813125444940, 5.89337862977893958941121052384, 6.36338767553654384138334354417, 7.38002146760642370323109342650, 8.046616878275771057379706890968
