| L(s) = 1 | − 0.311·2-s − 1.90·4-s + 2.68·7-s + 1.21·8-s + 0.214·11-s − 3.11·13-s − 0.836·14-s + 3.42·16-s − 0.474·17-s − 19-s − 0.0666·22-s − 2.47·23-s + 0.969·26-s − 5.11·28-s + 2.83·29-s − 4.90·31-s − 3.49·32-s + 0.147·34-s + 1.93·37-s + 0.311·38-s − 5.45·41-s + 10.7·43-s − 0.407·44-s + 0.769·46-s − 9.52·47-s + 0.230·49-s + 5.93·52-s + ⋯ |
| L(s) = 1 | − 0.219·2-s − 0.951·4-s + 1.01·7-s + 0.429·8-s + 0.0646·11-s − 0.864·13-s − 0.223·14-s + 0.857·16-s − 0.115·17-s − 0.229·19-s − 0.0142·22-s − 0.515·23-s + 0.190·26-s − 0.967·28-s + 0.526·29-s − 0.880·31-s − 0.617·32-s + 0.0253·34-s + 0.317·37-s + 0.0504·38-s − 0.852·41-s + 1.64·43-s − 0.0614·44-s + 0.113·46-s − 1.38·47-s + 0.0328·49-s + 0.822·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 0.311T + 2T^{2} \) |
| 7 | \( 1 - 2.68T + 7T^{2} \) |
| 11 | \( 1 - 0.214T + 11T^{2} \) |
| 13 | \( 1 + 3.11T + 13T^{2} \) |
| 17 | \( 1 + 0.474T + 17T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 2.83T + 29T^{2} \) |
| 31 | \( 1 + 4.90T + 31T^{2} \) |
| 37 | \( 1 - 1.93T + 37T^{2} \) |
| 41 | \( 1 + 5.45T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 9.52T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 6.56T + 59T^{2} \) |
| 61 | \( 1 - 7.76T + 61T^{2} \) |
| 67 | \( 1 + 6.85T + 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 16.6T + 73T^{2} \) |
| 79 | \( 1 - 1.37T + 79T^{2} \) |
| 83 | \( 1 + 1.65T + 83T^{2} \) |
| 89 | \( 1 + 16.5T + 89T^{2} \) |
| 97 | \( 1 - 9.05T + 97T^{2} \) |
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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
−8.102128579928093780819093489025, −7.55522140109057753411488453733, −6.63635463843122343156185068411, −5.60325001784764608487739305358, −4.93172357429566997633115711754, −4.40848465607322244136804202248, −3.52605344301102520749554057516, −2.30031809092734832419211003213, −1.31933166971218177959031336761, 0,
1.31933166971218177959031336761, 2.30031809092734832419211003213, 3.52605344301102520749554057516, 4.40848465607322244136804202248, 4.93172357429566997633115711754, 5.60325001784764608487739305358, 6.63635463843122343156185068411, 7.55522140109057753411488453733, 8.102128579928093780819093489025
