L(s) = 1 | + 2-s + 1.66·3-s + 4-s − 2.16·5-s + 1.66·6-s + 3.52·7-s + 8-s − 0.222·9-s − 2.16·10-s − 3.93·11-s + 1.66·12-s − 6.14·13-s + 3.52·14-s − 3.60·15-s + 16-s + 5.51·17-s − 0.222·18-s + 19-s − 2.16·20-s + 5.87·21-s − 3.93·22-s − 0.382·23-s + 1.66·24-s − 0.310·25-s − 6.14·26-s − 5.37·27-s + 3.52·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.962·3-s + 0.5·4-s − 0.968·5-s + 0.680·6-s + 1.33·7-s + 0.353·8-s − 0.0741·9-s − 0.684·10-s − 1.18·11-s + 0.481·12-s − 1.70·13-s + 0.942·14-s − 0.931·15-s + 0.250·16-s + 1.33·17-s − 0.0524·18-s + 0.229·19-s − 0.484·20-s + 1.28·21-s − 0.839·22-s − 0.0797·23-s + 0.340·24-s − 0.0621·25-s − 1.20·26-s − 1.03·27-s + 0.666·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 - 1.66T + 3T^{2} \) |
| 5 | \( 1 + 2.16T + 5T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 + 3.93T + 11T^{2} \) |
| 13 | \( 1 + 6.14T + 13T^{2} \) |
| 17 | \( 1 - 5.51T + 17T^{2} \) |
| 23 | \( 1 + 0.382T + 23T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 + 5.31T + 31T^{2} \) |
| 37 | \( 1 - 4.36T + 37T^{2} \) |
| 41 | \( 1 + 7.85T + 41T^{2} \) |
| 43 | \( 1 + 3.27T + 43T^{2} \) |
| 47 | \( 1 + 3.34T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 - 0.712T + 59T^{2} \) |
| 61 | \( 1 + 3.63T + 61T^{2} \) |
| 67 | \( 1 - 1.30T + 67T^{2} \) |
| 71 | \( 1 + 2.12T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 2.21T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 - 1.18T + 89T^{2} \) |
| 97 | \( 1 - 3.75T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.61176899266132268070903623243, −7.22407691865209094831430887818, −5.85165050449518062401826559786, −5.05529176863675809091399820066, −4.82351684869237956698001928995, −3.81364027766625603707903269600, −3.11575388998611551351779783055, −2.47778391316029153745985954122, −1.64077988517538465786514690808, 0,
1.64077988517538465786514690808, 2.47778391316029153745985954122, 3.11575388998611551351779783055, 3.81364027766625603707903269600, 4.82351684869237956698001928995, 5.05529176863675809091399820066, 5.85165050449518062401826559786, 7.22407691865209094831430887818, 7.61176899266132268070903623243