| L(s) = 1 | + 1.06·2-s − 0.326·3-s − 0.873·4-s − 3.24·5-s − 0.346·6-s + 7-s − 3.04·8-s − 2.89·9-s − 3.44·10-s − 3.51·11-s + 0.285·12-s + 3.47·13-s + 1.06·14-s + 1.06·15-s − 1.49·16-s − 4.60·17-s − 3.07·18-s + 4.37·19-s + 2.83·20-s − 0.326·21-s − 3.73·22-s − 1.10·23-s + 0.996·24-s + 5.56·25-s + 3.69·26-s + 1.92·27-s − 0.873·28-s + ⋯ |
| L(s) = 1 | + 0.750·2-s − 0.188·3-s − 0.436·4-s − 1.45·5-s − 0.141·6-s + 0.377·7-s − 1.07·8-s − 0.964·9-s − 1.09·10-s − 1.05·11-s + 0.0823·12-s + 0.965·13-s + 0.283·14-s + 0.274·15-s − 0.372·16-s − 1.11·17-s − 0.723·18-s + 1.00·19-s + 0.634·20-s − 0.0712·21-s − 0.795·22-s − 0.231·23-s + 0.203·24-s + 1.11·25-s + 0.724·26-s + 0.370·27-s − 0.165·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 301 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 1.06T + 2T^{2} \) |
| 3 | \( 1 + 0.326T + 3T^{2} \) |
| 5 | \( 1 + 3.24T + 5T^{2} \) |
| 11 | \( 1 + 3.51T + 11T^{2} \) |
| 13 | \( 1 - 3.47T + 13T^{2} \) |
| 17 | \( 1 + 4.60T + 17T^{2} \) |
| 19 | \( 1 - 4.37T + 19T^{2} \) |
| 23 | \( 1 + 1.10T + 23T^{2} \) |
| 29 | \( 1 + 7.18T + 29T^{2} \) |
| 31 | \( 1 - 1.25T + 31T^{2} \) |
| 37 | \( 1 + 7.20T + 37T^{2} \) |
| 41 | \( 1 - 8.91T + 41T^{2} \) |
| 47 | \( 1 + 2.54T + 47T^{2} \) |
| 53 | \( 1 + 0.312T + 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 2.59T + 61T^{2} \) |
| 67 | \( 1 + 8.46T + 67T^{2} \) |
| 71 | \( 1 - 8.92T + 71T^{2} \) |
| 73 | \( 1 + 8.68T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 1.64T + 83T^{2} \) |
| 89 | \( 1 - 6.82T + 89T^{2} \) |
| 97 | \( 1 - 0.914T + 97T^{2} \) |
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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
−11.40796677997079829953232460894, −10.80510964399719611107674759292, −9.117510192574497393095227455867, −8.351797377646347940488861806143, −7.51724631548305982638459039112, −5.98740429167446979125463969639, −5.04713632479100961847997212421, −4.03445856088550102793791588746, −3.03962472791435211558083694504, 0,
3.03962472791435211558083694504, 4.03445856088550102793791588746, 5.04713632479100961847997212421, 5.98740429167446979125463969639, 7.51724631548305982638459039112, 8.351797377646347940488861806143, 9.117510192574497393095227455867, 10.80510964399719611107674759292, 11.40796677997079829953232460894
