| L(s) = 1 | + 3-s − 2·4-s + 7-s + 9-s − 2·12-s + 4·16-s + 6·17-s − 7·19-s + 21-s − 6·23-s + 27-s − 2·28-s − 6·29-s + 5·31-s − 2·36-s + 2·37-s + 6·41-s − 4·43-s + 6·47-s + 4·48-s + 49-s + 6·51-s + 12·53-s − 7·57-s − 6·59-s + 11·61-s + 63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s − 0.577·12-s + 16-s + 1.45·17-s − 1.60·19-s + 0.218·21-s − 1.25·23-s + 0.192·27-s − 0.377·28-s − 1.11·29-s + 0.898·31-s − 1/3·36-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.875·47-s + 0.577·48-s + 1/7·49-s + 0.840·51-s + 1.64·53-s − 0.927·57-s − 0.781·59-s + 1.40·61-s + 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.263813476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.263813476\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−13.97444620697645, −13.38541055058528, −13.01974742822682, −12.45635521610636, −12.07255137643126, −11.51174419324513, −10.70853736113887, −10.28304707504073, −9.917383977131550, −9.316685073096737, −8.881648127298750, −8.255920050249774, −8.009036445175510, −7.540542716466743, −6.802080410294551, −6.018549854577693, −5.667790584234686, −5.003362360252010, −4.371104351305916, −3.879474574728974, −3.556058525046800, −2.580006134562731, −2.071332467881381, −1.226398646873245, −0.5066333097081022,
0.5066333097081022, 1.226398646873245, 2.071332467881381, 2.580006134562731, 3.556058525046800, 3.879474574728974, 4.371104351305916, 5.003362360252010, 5.667790584234686, 6.018549854577693, 6.802080410294551, 7.540542716466743, 8.009036445175510, 8.255920050249774, 8.881648127298750, 9.316685073096737, 9.917383977131550, 10.28304707504073, 10.70853736113887, 11.51174419324513, 12.07255137643126, 12.45635521610636, 13.01974742822682, 13.38541055058528, 13.97444620697645
