| L(s) = 1 | + 2-s + 4-s − 3·5-s + 8-s − 3·10-s + 4·13-s + 16-s − 6·17-s + 4·19-s − 3·20-s + 6·23-s + 4·25-s + 4·26-s − 3·29-s − 8·31-s + 32-s − 6·34-s + 8·37-s + 4·38-s − 3·40-s + 6·41-s + 8·43-s + 6·46-s − 6·47-s + 4·50-s + 4·52-s + 9·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.353·8-s − 0.948·10-s + 1.10·13-s + 1/4·16-s − 1.45·17-s + 0.917·19-s − 0.670·20-s + 1.25·23-s + 4/5·25-s + 0.784·26-s − 0.557·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s + 1.31·37-s + 0.648·38-s − 0.474·40-s + 0.937·41-s + 1.21·43-s + 0.884·46-s − 0.875·47-s + 0.565·50-s + 0.554·52-s + 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.244083894\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.244083894\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.825535594064392160521257380626, −7.911602813414596687279622029152, −7.32165843000362378045556900335, −6.60530141529327079228089604135, −5.68600195051065591975234189108, −4.79175091319454268199241885671, −3.96202337236382740315337262000, −3.48497373275609492712736133844, −2.35417663398115400765507162847, −0.854350917741434894530487190990,
0.854350917741434894530487190990, 2.35417663398115400765507162847, 3.48497373275609492712736133844, 3.96202337236382740315337262000, 4.79175091319454268199241885671, 5.68600195051065591975234189108, 6.60530141529327079228089604135, 7.32165843000362378045556900335, 7.911602813414596687279622029152, 8.825535594064392160521257380626
