| L(s) = 1 | + 2.01·2-s + 2.07·4-s + 1.01·7-s + 0.153·8-s − 4.75·11-s − 0.103·13-s + 2.05·14-s − 3.84·16-s + 5.83·17-s − 0.724·19-s − 9.60·22-s + 9.07·23-s − 0.209·26-s + 2.11·28-s + 3.98·29-s + 1.06·31-s − 8.06·32-s + 11.7·34-s + 4.02·37-s − 1.46·38-s − 7.20·41-s + 8.62·43-s − 9.88·44-s + 18.3·46-s + 8.19·47-s − 5.96·49-s − 0.215·52-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 1.03·4-s + 0.385·7-s + 0.0541·8-s − 1.43·11-s − 0.0287·13-s + 0.549·14-s − 0.960·16-s + 1.41·17-s − 0.166·19-s − 2.04·22-s + 1.89·23-s − 0.0411·26-s + 0.399·28-s + 0.740·29-s + 0.191·31-s − 1.42·32-s + 2.02·34-s + 0.661·37-s − 0.237·38-s − 1.12·41-s + 1.31·43-s − 1.48·44-s + 2.70·46-s + 1.19·47-s − 0.851·49-s − 0.0298·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5625 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.282514523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.282514523\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 2.01T + 2T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 + 4.75T + 11T^{2} \) |
| 13 | \( 1 + 0.103T + 13T^{2} \) |
| 17 | \( 1 - 5.83T + 17T^{2} \) |
| 19 | \( 1 + 0.724T + 19T^{2} \) |
| 23 | \( 1 - 9.07T + 23T^{2} \) |
| 29 | \( 1 - 3.98T + 29T^{2} \) |
| 31 | \( 1 - 1.06T + 31T^{2} \) |
| 37 | \( 1 - 4.02T + 37T^{2} \) |
| 41 | \( 1 + 7.20T + 41T^{2} \) |
| 43 | \( 1 - 8.62T + 43T^{2} \) |
| 47 | \( 1 - 8.19T + 47T^{2} \) |
| 53 | \( 1 - 4.36T + 53T^{2} \) |
| 59 | \( 1 - 4.91T + 59T^{2} \) |
| 61 | \( 1 - 6.96T + 61T^{2} \) |
| 67 | \( 1 + 9.91T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 8.63T + 73T^{2} \) |
| 79 | \( 1 - 2.48T + 79T^{2} \) |
| 83 | \( 1 + 4.24T + 83T^{2} \) |
| 89 | \( 1 - 18.3T + 89T^{2} \) |
| 97 | \( 1 - 6.69T + 97T^{2} \) |
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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
−7.951205907658832556090230582891, −7.31289635820110720392112623007, −6.53568171323824847067673869789, −5.63854357994867101076756776339, −5.18100069041921323436926449450, −4.68890302667939658784449378378, −3.69879737679564379824805581532, −2.95170825564703598804916283713, −2.36242749181773060006634961586, −0.893698584710080695866631914176,
0.893698584710080695866631914176, 2.36242749181773060006634961586, 2.95170825564703598804916283713, 3.69879737679564379824805581532, 4.68890302667939658784449378378, 5.18100069041921323436926449450, 5.63854357994867101076756776339, 6.53568171323824847067673869789, 7.31289635820110720392112623007, 7.951205907658832556090230582891
