| L(s) = 1 | + 1.59·2-s + 0.133·3-s + 0.536·4-s + 3.85·5-s + 0.211·6-s − 3.78·7-s − 2.33·8-s − 2.98·9-s + 6.13·10-s + 3.09·11-s + 0.0714·12-s + 5.35·13-s − 6.02·14-s + 0.512·15-s − 4.78·16-s + 4.63·17-s − 4.74·18-s + 7.98·19-s + 2.06·20-s − 0.503·21-s + 4.93·22-s − 7.87·23-s − 0.310·24-s + 9.83·25-s + 8.53·26-s − 0.796·27-s − 2.02·28-s + ⋯ |
| L(s) = 1 | + 1.12·2-s + 0.0768·3-s + 0.268·4-s + 1.72·5-s + 0.0865·6-s − 1.42·7-s − 0.824·8-s − 0.994·9-s + 1.94·10-s + 0.933·11-s + 0.0206·12-s + 1.48·13-s − 1.61·14-s + 0.132·15-s − 1.19·16-s + 1.12·17-s − 1.11·18-s + 1.83·19-s + 0.462·20-s − 0.109·21-s + 1.05·22-s − 1.64·23-s − 0.0633·24-s + 1.96·25-s + 1.67·26-s − 0.153·27-s − 0.383·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.827191164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.827191164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2671 | \( 1 - T \) |
| good | 2 | \( 1 - 1.59T + 2T^{2} \) |
| 3 | \( 1 - 0.133T + 3T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 7 | \( 1 + 3.78T + 7T^{2} \) |
| 11 | \( 1 - 3.09T + 11T^{2} \) |
| 13 | \( 1 - 5.35T + 13T^{2} \) |
| 17 | \( 1 - 4.63T + 17T^{2} \) |
| 19 | \( 1 - 7.98T + 19T^{2} \) |
| 23 | \( 1 + 7.87T + 23T^{2} \) |
| 29 | \( 1 - 4.15T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 + 4.72T + 37T^{2} \) |
| 41 | \( 1 - 7.47T + 41T^{2} \) |
| 43 | \( 1 + 9.48T + 43T^{2} \) |
| 47 | \( 1 - 6.78T + 47T^{2} \) |
| 53 | \( 1 + 1.40T + 53T^{2} \) |
| 59 | \( 1 - 6.62T + 59T^{2} \) |
| 61 | \( 1 + 0.305T + 61T^{2} \) |
| 67 | \( 1 - 13.7T + 67T^{2} \) |
| 71 | \( 1 - 9.58T + 71T^{2} \) |
| 73 | \( 1 - 5.49T + 73T^{2} \) |
| 79 | \( 1 + 8.56T + 79T^{2} \) |
| 83 | \( 1 - 7.47T + 83T^{2} \) |
| 89 | \( 1 + 5.28T + 89T^{2} \) |
| 97 | \( 1 + 8.88T + 97T^{2} \) |
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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
−9.042710488267943391701498554819, −8.281147829990642710403977251878, −6.72094418799550302128396037124, −6.26000158773105364359942535371, −5.70875335800584157372388019165, −5.32772020399234906420657181766, −3.77505173548663104403227956726, −3.34324728864091782569897096132, −2.51040965122619974355802510118, −1.09487678465242126670742105357,
1.09487678465242126670742105357, 2.51040965122619974355802510118, 3.34324728864091782569897096132, 3.77505173548663104403227956726, 5.32772020399234906420657181766, 5.70875335800584157372388019165, 6.26000158773105364359942535371, 6.72094418799550302128396037124, 8.281147829990642710403977251878, 9.042710488267943391701498554819
