| L(s) = 1 | − 1.63·2-s + 0.960·3-s + 0.686·4-s + 5-s − 1.57·6-s + 4.58·7-s + 2.15·8-s − 2.07·9-s − 1.63·10-s − 3.05·11-s + 0.659·12-s − 2.91·13-s − 7.50·14-s + 0.960·15-s − 4.90·16-s − 17-s + 3.40·18-s − 5.62·19-s + 0.686·20-s + 4.40·21-s + 5.00·22-s + 1.25·23-s + 2.06·24-s + 25-s + 4.77·26-s − 4.87·27-s + 3.14·28-s + ⋯ |
| L(s) = 1 | − 1.15·2-s + 0.554·3-s + 0.343·4-s + 0.447·5-s − 0.642·6-s + 1.73·7-s + 0.761·8-s − 0.692·9-s − 0.518·10-s − 0.920·11-s + 0.190·12-s − 0.807·13-s − 2.00·14-s + 0.248·15-s − 1.22·16-s − 0.242·17-s + 0.802·18-s − 1.28·19-s + 0.153·20-s + 0.960·21-s + 1.06·22-s + 0.261·23-s + 0.422·24-s + 0.200·25-s + 0.935·26-s − 0.938·27-s + 0.594·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 + 1.63T + 2T^{2} \) |
| 3 | \( 1 - 0.960T + 3T^{2} \) |
| 7 | \( 1 - 4.58T + 7T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 + 2.91T + 13T^{2} \) |
| 19 | \( 1 + 5.62T + 19T^{2} \) |
| 23 | \( 1 - 1.25T + 23T^{2} \) |
| 29 | \( 1 + 1.71T + 29T^{2} \) |
| 31 | \( 1 - 8.18T + 31T^{2} \) |
| 37 | \( 1 - 3.67T + 37T^{2} \) |
| 41 | \( 1 - 12.2T + 41T^{2} \) |
| 43 | \( 1 + 9.45T + 43T^{2} \) |
| 47 | \( 1 + 0.692T + 47T^{2} \) |
| 53 | \( 1 + 3.94T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 9.57T + 61T^{2} \) |
| 67 | \( 1 - 5.76T + 67T^{2} \) |
| 73 | \( 1 + 1.79T + 73T^{2} \) |
| 79 | \( 1 - 0.429T + 79T^{2} \) |
| 83 | \( 1 - 4.91T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 3.64T + 97T^{2} \) |
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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
−8.031702557751779627584710586033, −7.50358595678956284292706995572, −6.49743376494220543268143171151, −5.48152522678779418096164454365, −4.79529907938765544506554453251, −4.26587518683091230432351710399, −2.66899361672629377253342375263, −2.23581985116145329036984458561, −1.33075456377322554649162431753, 0,
1.33075456377322554649162431753, 2.23581985116145329036984458561, 2.66899361672629377253342375263, 4.26587518683091230432351710399, 4.79529907938765544506554453251, 5.48152522678779418096164454365, 6.49743376494220543268143171151, 7.50358595678956284292706995572, 8.031702557751779627584710586033
