| L(s) = 1 | + 1.07·3-s + 1.81·5-s + 7-s − 1.84·9-s + 4.90·11-s + 4.83·13-s + 1.94·15-s + 17-s + 5.27·19-s + 1.07·21-s − 3.94·23-s − 1.71·25-s − 5.20·27-s + 2.75·29-s − 1.28·31-s + 5.27·33-s + 1.81·35-s − 0.798·37-s + 5.18·39-s + 4.50·41-s − 5.71·43-s − 3.34·45-s + 3.76·47-s + 49-s + 1.07·51-s + 7.86·53-s + 8.90·55-s + ⋯ |
| L(s) = 1 | + 0.619·3-s + 0.810·5-s + 0.377·7-s − 0.615·9-s + 1.48·11-s + 1.34·13-s + 0.502·15-s + 0.242·17-s + 1.20·19-s + 0.234·21-s − 0.822·23-s − 0.342·25-s − 1.00·27-s + 0.512·29-s − 0.230·31-s + 0.917·33-s + 0.306·35-s − 0.131·37-s + 0.830·39-s + 0.704·41-s − 0.871·43-s − 0.499·45-s + 0.549·47-s + 0.142·49-s + 0.150·51-s + 1.08·53-s + 1.20·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.829784536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.829784536\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 1.07T + 3T^{2} \) |
| 5 | \( 1 - 1.81T + 5T^{2} \) |
| 11 | \( 1 - 4.90T + 11T^{2} \) |
| 13 | \( 1 - 4.83T + 13T^{2} \) |
| 19 | \( 1 - 5.27T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 - 2.75T + 29T^{2} \) |
| 31 | \( 1 + 1.28T + 31T^{2} \) |
| 37 | \( 1 + 0.798T + 37T^{2} \) |
| 41 | \( 1 - 4.50T + 41T^{2} \) |
| 43 | \( 1 + 5.71T + 43T^{2} \) |
| 47 | \( 1 - 3.76T + 47T^{2} \) |
| 53 | \( 1 - 7.86T + 53T^{2} \) |
| 59 | \( 1 + 0.722T + 59T^{2} \) |
| 61 | \( 1 - 8.29T + 61T^{2} \) |
| 67 | \( 1 + 0.598T + 67T^{2} \) |
| 71 | \( 1 - 15.1T + 71T^{2} \) |
| 73 | \( 1 + 7.53T + 73T^{2} \) |
| 79 | \( 1 - 0.156T + 79T^{2} \) |
| 83 | \( 1 - 6.38T + 83T^{2} \) |
| 89 | \( 1 + 0.304T + 89T^{2} \) |
| 97 | \( 1 + 16.5T + 97T^{2} \) |
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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
−8.132215679823603422358637364130, −7.17026405176123126212638242666, −6.38915267055710032647135228565, −5.82277027233187703610923638505, −5.26555225073943263990583540058, −3.99542896231249574997407201856, −3.64374770573307582821264899496, −2.65652683812565069284549930518, −1.75247530431144299956767532799, −1.04125168637422288080527696755,
1.04125168637422288080527696755, 1.75247530431144299956767532799, 2.65652683812565069284549930518, 3.64374770573307582821264899496, 3.99542896231249574997407201856, 5.26555225073943263990583540058, 5.82277027233187703610923638505, 6.38915267055710032647135228565, 7.17026405176123126212638242666, 8.132215679823603422358637364130
