| L(s) = 1 | − 2.19·2-s + 1.08·3-s + 2.82·4-s − 3.90·5-s − 2.39·6-s − 0.904·7-s − 1.80·8-s − 1.81·9-s + 8.56·10-s + 0.774·11-s + 3.07·12-s + 3.58·13-s + 1.98·14-s − 4.24·15-s − 1.68·16-s − 0.236·17-s + 3.98·18-s − 6.17·19-s − 11.0·20-s − 0.985·21-s − 1.70·22-s + 6.36·23-s − 1.96·24-s + 10.2·25-s − 7.86·26-s − 5.24·27-s − 2.55·28-s + ⋯ |
| L(s) = 1 | − 1.55·2-s + 0.628·3-s + 1.41·4-s − 1.74·5-s − 0.976·6-s − 0.341·7-s − 0.637·8-s − 0.604·9-s + 2.70·10-s + 0.233·11-s + 0.886·12-s + 0.993·13-s + 0.530·14-s − 1.09·15-s − 0.420·16-s − 0.0573·17-s + 0.938·18-s − 1.41·19-s − 2.46·20-s − 0.215·21-s − 0.362·22-s + 1.32·23-s − 0.400·24-s + 2.04·25-s − 1.54·26-s − 1.00·27-s − 0.482·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9409 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5328759811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5328759811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 97 | \( 1 \) |
| good | 2 | \( 1 + 2.19T + 2T^{2} \) |
| 3 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 + 3.90T + 5T^{2} \) |
| 7 | \( 1 + 0.904T + 7T^{2} \) |
| 11 | \( 1 - 0.774T + 11T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 + 0.236T + 17T^{2} \) |
| 19 | \( 1 + 6.17T + 19T^{2} \) |
| 23 | \( 1 - 6.36T + 23T^{2} \) |
| 29 | \( 1 - 7.97T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 - 6.95T + 37T^{2} \) |
| 41 | \( 1 - 9.75T + 41T^{2} \) |
| 43 | \( 1 - 7.36T + 43T^{2} \) |
| 47 | \( 1 + 2.70T + 47T^{2} \) |
| 53 | \( 1 + 9.75T + 53T^{2} \) |
| 59 | \( 1 + 3.22T + 59T^{2} \) |
| 61 | \( 1 + 6.88T + 61T^{2} \) |
| 67 | \( 1 + 2.46T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 5.33T + 73T^{2} \) |
| 79 | \( 1 + 16.2T + 79T^{2} \) |
| 83 | \( 1 - 3.39T + 83T^{2} \) |
| 89 | \( 1 - 0.220T + 89T^{2} \) |
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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
−7.87668127027006160866703881846, −7.48481371500359301668826495915, −6.56235824089746830727806955075, −6.14291359568230024558098313886, −4.55759431641679757670355903858, −4.19505784436881048782198497448, −3.08525786807537393647563746927, −2.73258180606919328329376467711, −1.33760899294571311563401127501, −0.46785966812109298275237137539,
0.46785966812109298275237137539, 1.33760899294571311563401127501, 2.73258180606919328329376467711, 3.08525786807537393647563746927, 4.19505784436881048782198497448, 4.55759431641679757670355903858, 6.14291359568230024558098313886, 6.56235824089746830727806955075, 7.48481371500359301668826495915, 7.87668127027006160866703881846
