L(s) = 1 | − 2.55·2-s − 3.23·3-s + 4.52·4-s − 3.35·5-s + 8.26·6-s + 1.37·7-s − 6.44·8-s + 7.46·9-s + 8.57·10-s + 11-s − 14.6·12-s − 5.93·13-s − 3.52·14-s + 10.8·15-s + 7.41·16-s + 17-s − 19.0·18-s + 0.524·19-s − 15.1·20-s − 4.45·21-s − 2.55·22-s + 5.85·23-s + 20.8·24-s + 6.26·25-s + 15.1·26-s − 14.4·27-s + 6.23·28-s + ⋯ |
L(s) = 1 | − 1.80·2-s − 1.86·3-s + 2.26·4-s − 1.50·5-s + 3.37·6-s + 0.520·7-s − 2.27·8-s + 2.48·9-s + 2.71·10-s + 0.301·11-s − 4.22·12-s − 1.64·13-s − 0.940·14-s + 2.80·15-s + 1.85·16-s + 0.242·17-s − 4.49·18-s + 0.120·19-s − 3.39·20-s − 0.972·21-s − 0.544·22-s + 1.22·23-s + 4.25·24-s + 1.25·25-s + 2.97·26-s − 2.77·27-s + 1.17·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1713335338\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1713335338\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 + 2.55T + 2T^{2} \) |
| 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 + 3.35T + 5T^{2} \) |
| 7 | \( 1 - 1.37T + 7T^{2} \) |
| 13 | \( 1 + 5.93T + 13T^{2} \) |
| 19 | \( 1 - 0.524T + 19T^{2} \) |
| 23 | \( 1 - 5.85T + 23T^{2} \) |
| 29 | \( 1 + 0.806T + 29T^{2} \) |
| 31 | \( 1 + 8.85T + 31T^{2} \) |
| 37 | \( 1 + 1.29T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 47 | \( 1 + 9.73T + 47T^{2} \) |
| 53 | \( 1 - 5.69T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 + 2.19T + 61T^{2} \) |
| 67 | \( 1 - 2.89T + 67T^{2} \) |
| 71 | \( 1 - 8.86T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 1.57T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−7.51542508333476247506585102955, −7.36145246201145907279049633847, −6.89787372327868220931250617538, −5.96251315816606924255299643169, −5.06991052123563867742557396844, −4.55842492587115179841031553852, −3.49283408931967250149979489218, −2.15658406221455306172909243716, −1.09197733153475773855176779920, −0.36831469599130654048141629347,
0.36831469599130654048141629347, 1.09197733153475773855176779920, 2.15658406221455306172909243716, 3.49283408931967250149979489218, 4.55842492587115179841031553852, 5.06991052123563867742557396844, 5.96251315816606924255299643169, 6.89787372327868220931250617538, 7.36145246201145907279049633847, 7.51542508333476247506585102955