| L(s) = 1 | − 3-s − 1.78·5-s − 2.07·7-s + 9-s + 2.20·11-s − 6.61·13-s + 1.78·15-s − 6.93·17-s − 3.40·19-s + 2.07·21-s + 2.31·23-s − 1.81·25-s − 27-s − 1.77·29-s + 5.22·31-s − 2.20·33-s + 3.71·35-s − 10.8·37-s + 6.61·39-s + 3.10·41-s − 3.95·43-s − 1.78·45-s + 6.71·47-s − 2.67·49-s + 6.93·51-s − 9.33·53-s − 3.93·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.798·5-s − 0.785·7-s + 0.333·9-s + 0.664·11-s − 1.83·13-s + 0.461·15-s − 1.68·17-s − 0.781·19-s + 0.453·21-s + 0.483·23-s − 0.362·25-s − 0.192·27-s − 0.329·29-s + 0.938·31-s − 0.383·33-s + 0.627·35-s − 1.78·37-s + 1.05·39-s + 0.485·41-s − 0.603·43-s − 0.266·45-s + 0.978·47-s − 0.382·49-s + 0.971·51-s − 1.28·53-s − 0.530·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2557355789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2557355789\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 503 | \( 1 + T \) |
| good | 5 | \( 1 + 1.78T + 5T^{2} \) |
| 7 | \( 1 + 2.07T + 7T^{2} \) |
| 11 | \( 1 - 2.20T + 11T^{2} \) |
| 13 | \( 1 + 6.61T + 13T^{2} \) |
| 17 | \( 1 + 6.93T + 17T^{2} \) |
| 19 | \( 1 + 3.40T + 19T^{2} \) |
| 23 | \( 1 - 2.31T + 23T^{2} \) |
| 29 | \( 1 + 1.77T + 29T^{2} \) |
| 31 | \( 1 - 5.22T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 3.10T + 41T^{2} \) |
| 43 | \( 1 + 3.95T + 43T^{2} \) |
| 47 | \( 1 - 6.71T + 47T^{2} \) |
| 53 | \( 1 + 9.33T + 53T^{2} \) |
| 59 | \( 1 + 8.55T + 59T^{2} \) |
| 61 | \( 1 - 8.78T + 61T^{2} \) |
| 67 | \( 1 + 8.71T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 9.60T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 3.56T + 83T^{2} \) |
| 89 | \( 1 - 11.0T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.924602644873091374368669120691, −7.22397036082009629567655370132, −6.67949149978608698410697313485, −6.16431771472366048600713327562, −4.99602051170556647505776540639, −4.52127099770632993391397221519, −3.78279911659493265783682977152, −2.80017058462752166558248816448, −1.87318499919081933578392483283, −0.25799222473851706727821619090,
0.25799222473851706727821619090, 1.87318499919081933578392483283, 2.80017058462752166558248816448, 3.78279911659493265783682977152, 4.52127099770632993391397221519, 4.99602051170556647505776540639, 6.16431771472366048600713327562, 6.67949149978608698410697313485, 7.22397036082009629567655370132, 7.924602644873091374368669120691
