| L(s) = 1 | + 1.41·2-s − 1.41·3-s − 3.41·5-s − 2.00·6-s + 3.41·7-s − 2.82·8-s − 0.999·9-s − 4.82·10-s − 3.82·11-s − 1.82·13-s + 4.82·14-s + 4.82·15-s − 4.00·16-s + 2.17·17-s − 1.41·18-s − 0.828·19-s − 4.82·21-s − 5.41·22-s + 6.65·23-s + 4·24-s + 6.65·25-s − 2.58·26-s + 5.65·27-s + 4.24·29-s + 6.82·30-s + ⋯ |
| L(s) = 1 | + 1.00·2-s − 0.816·3-s − 1.52·5-s − 0.816·6-s + 1.29·7-s − 0.999·8-s − 0.333·9-s − 1.52·10-s − 1.15·11-s − 0.507·13-s + 1.29·14-s + 1.24·15-s − 1.00·16-s + 0.526·17-s − 0.333·18-s − 0.190·19-s − 1.05·21-s − 1.15·22-s + 1.38·23-s + 0.816·24-s + 1.33·25-s − 0.507·26-s + 1.08·27-s + 0.787·29-s + 1.24·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.038873543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.038873543\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.41T + 2T^{2} \) |
| 3 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 + 3.41T + 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 + 3.82T + 11T^{2} \) |
| 13 | \( 1 + 1.82T + 13T^{2} \) |
| 17 | \( 1 - 2.17T + 17T^{2} \) |
| 19 | \( 1 + 0.828T + 19T^{2} \) |
| 23 | \( 1 - 6.65T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 13.8T + 53T^{2} \) |
| 59 | \( 1 + 4.82T + 59T^{2} \) |
| 61 | \( 1 - 0.242T + 61T^{2} \) |
| 67 | \( 1 + 7.48T + 67T^{2} \) |
| 71 | \( 1 - 3.17T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 4.82T + 79T^{2} \) |
| 83 | \( 1 - 3.34T + 83T^{2} \) |
| 89 | \( 1 - 1.75T + 89T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
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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
−8.973916377641076040059756235198, −8.320586783088866980162415011314, −7.64146939619317867109834206620, −6.83563376666240588720467186343, −5.54293668377903179308609194785, −5.07961553019777875875038953922, −4.58413157866908125296916523784, −3.58427047696094590861953772434, −2.63969668236081009896094900574, −0.60206100758918104557594532943,
0.60206100758918104557594532943, 2.63969668236081009896094900574, 3.58427047696094590861953772434, 4.58413157866908125296916523784, 5.07961553019777875875038953922, 5.54293668377903179308609194785, 6.83563376666240588720467186343, 7.64146939619317867109834206620, 8.320586783088866980162415011314, 8.973916377641076040059756235198
