| L(s) = 1 | + 2·2-s + 4·4-s − 5·5-s + 7·7-s + 8·8-s + 9·9-s − 10·10-s + 14·14-s + 16·16-s − 6·17-s + 18·18-s + 18·19-s − 20·20-s + 25·25-s + 28·28-s + 32·32-s − 12·34-s − 35·35-s + 36·36-s − 66·37-s + 36·38-s − 40·40-s − 54·43-s − 45·45-s − 66·47-s + 49·49-s + 50·50-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s + 9-s − 10-s + 14-s + 16-s − 0.352·17-s + 18-s + 0.947·19-s − 20-s + 25-s + 28-s + 32-s − 0.352·34-s − 35-s + 36-s − 1.78·37-s + 0.947·38-s − 40-s − 1.25·43-s − 45-s − 1.40·47-s + 49-s + 50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.043811824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.043811824\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( ( 1 - p T )( 1 + p T ) \) |
| 17 | \( 1 + 6 T + p^{2} T^{2} \) |
| 19 | \( 1 - 18 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 66 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 54 T + p^{2} T^{2} \) |
| 47 | \( 1 + 66 T + p^{2} T^{2} \) |
| 53 | \( 1 + 34 T + p^{2} T^{2} \) |
| 59 | \( 1 + 62 T + p^{2} T^{2} \) |
| 61 | \( 1 - 102 T + p^{2} T^{2} \) |
| 67 | \( 1 + 6 T + p^{2} T^{2} \) |
| 71 | \( 1 + 138 T + p^{2} T^{2} \) |
| 73 | \( 1 - 106 T + p^{2} T^{2} \) |
| 79 | \( 1 + 122 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 + 166 T + p^{2} T^{2} \) |
| 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
−11.73429140961958786930640840323, −11.06086493016575800753374540858, −10.06416078141395632984670919850, −8.441969840201049063515296434506, −7.51882429099875225012008956142, −6.80358192359358670525076178068, −5.19310709278045772391672071828, −4.44223524308516937445764357983, −3.36764394621950807431395684351, −1.58056477560737555483162335473,
1.58056477560737555483162335473, 3.36764394621950807431395684351, 4.44223524308516937445764357983, 5.19310709278045772391672071828, 6.80358192359358670525076178068, 7.51882429099875225012008956142, 8.441969840201049063515296434506, 10.06416078141395632984670919850, 11.06086493016575800753374540858, 11.73429140961958786930640840323
