| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s + 9-s + 12-s + 2·13-s − 14-s + 16-s + 6·17-s + 18-s + 5·19-s − 21-s + 24-s − 5·25-s + 2·26-s + 27-s − 28-s + 6·29-s − 4·31-s + 32-s + 6·34-s + 36-s + 8·37-s + 5·38-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.14·19-s − 0.218·21-s + 0.204·24-s − 25-s + 0.392·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 1.31·37-s + 0.811·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.908232669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.908232669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 127 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−10.07850886672061059774887164143, −9.763150850055314268514265511614, −8.456030467515693104967547882689, −7.74609743604347874728704384536, −6.78860582496417514402875220589, −5.83169729985955676404261763895, −4.90413220802049523654355402855, −3.61371821867750681081884149864, −3.06109345009375161162336387256, −1.50677993660887870810617642251,
1.50677993660887870810617642251, 3.06109345009375161162336387256, 3.61371821867750681081884149864, 4.90413220802049523654355402855, 5.83169729985955676404261763895, 6.78860582496417514402875220589, 7.74609743604347874728704384536, 8.456030467515693104967547882689, 9.763150850055314268514265511614, 10.07850886672061059774887164143
