| L(s) = 1 | + 1.63·2-s − 1.16·3-s + 0.660·4-s + 1.75·5-s − 1.90·6-s + 5.06·7-s − 2.18·8-s − 1.63·9-s + 2.86·10-s + 1.08·11-s − 0.771·12-s + 3.01·13-s + 8.25·14-s − 2.04·15-s − 4.88·16-s + 2.47·17-s − 2.66·18-s − 7.12·19-s + 1.15·20-s − 5.91·21-s + 1.76·22-s + 5.33·23-s + 2.55·24-s − 1.92·25-s + 4.91·26-s + 5.41·27-s + 3.34·28-s + ⋯ |
| L(s) = 1 | + 1.15·2-s − 0.674·3-s + 0.330·4-s + 0.784·5-s − 0.777·6-s + 1.91·7-s − 0.772·8-s − 0.545·9-s + 0.904·10-s + 0.325·11-s − 0.222·12-s + 0.835·13-s + 2.20·14-s − 0.529·15-s − 1.22·16-s + 0.600·17-s − 0.629·18-s − 1.63·19-s + 0.259·20-s − 1.28·21-s + 0.376·22-s + 1.11·23-s + 0.520·24-s − 0.384·25-s + 0.963·26-s + 1.04·27-s + 0.632·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 241 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.001472496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.001472496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 241 | \( 1 - T \) |
| good | 2 | \( 1 - 1.63T + 2T^{2} \) |
| 3 | \( 1 + 1.16T + 3T^{2} \) |
| 5 | \( 1 - 1.75T + 5T^{2} \) |
| 7 | \( 1 - 5.06T + 7T^{2} \) |
| 11 | \( 1 - 1.08T + 11T^{2} \) |
| 13 | \( 1 - 3.01T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 + 7.12T + 19T^{2} \) |
| 23 | \( 1 - 5.33T + 23T^{2} \) |
| 29 | \( 1 + 6.80T + 29T^{2} \) |
| 31 | \( 1 + 8.37T + 31T^{2} \) |
| 37 | \( 1 + 2.09T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 + 5.49T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 + 4.04T + 53T^{2} \) |
| 59 | \( 1 - 5.64T + 59T^{2} \) |
| 61 | \( 1 + 2.03T + 61T^{2} \) |
| 67 | \( 1 - 7.65T + 67T^{2} \) |
| 71 | \( 1 - 12.2T + 71T^{2} \) |
| 73 | \( 1 - 0.733T + 73T^{2} \) |
| 79 | \( 1 - 6.86T + 79T^{2} \) |
| 83 | \( 1 - 5.58T + 83T^{2} \) |
| 89 | \( 1 - 9.62T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−12.13578200079549243339189935132, −11.26197790127079380058659074650, −10.78087695032277116816593038871, −9.082569623199201025048022172020, −8.271795502667768229441384096974, −6.61334295014171908848321950373, −5.52800771018599377657892065584, −5.12363613374426988957280647962, −3.82380748007777464558800855376, −1.90393363054433790287308714865,
1.90393363054433790287308714865, 3.82380748007777464558800855376, 5.12363613374426988957280647962, 5.52800771018599377657892065584, 6.61334295014171908848321950373, 8.271795502667768229441384096974, 9.082569623199201025048022172020, 10.78087695032277116816593038871, 11.26197790127079380058659074650, 12.13578200079549243339189935132
