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
\(L(1)\) |
\(\approx\) |
\(0.2140371521\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2140371521\) |
\(L(\frac{3}{2})\) |
|
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
−8.291663089369118723276607919652, −7.75498571426441115952170904939, −6.74008252144289414785470668438, −6.44873035176682306068245891843, −5.24036555061593347705633360635, −4.58198061543401459934321434748, −3.49477456506453883719513429316, −2.39489810737147324306282910965, −1.81642737932227625191335261394, −0.29046410226323060869455092292,
0.29046410226323060869455092292, 1.81642737932227625191335261394, 2.39489810737147324306282910965, 3.49477456506453883719513429316, 4.58198061543401459934321434748, 5.24036555061593347705633360635, 6.44873035176682306068245891843, 6.74008252144289414785470668438, 7.75498571426441115952170904939, 8.291663089369118723276607919652