| L(s) = 1 | − 2.25·3-s − 5-s − 0.859·7-s + 2.07·9-s − 2.82·11-s + 1.99·13-s + 2.25·15-s + 5.17·17-s − 5.67·19-s + 1.93·21-s + 3.79·23-s + 25-s + 2.07·27-s − 6.51·29-s + 7.37·31-s + 6.37·33-s + 0.859·35-s + 1.97·37-s − 4.48·39-s + 0.390·41-s − 5.73·43-s − 2.07·45-s − 0.312·47-s − 6.26·49-s − 11.6·51-s − 7.73·53-s + 2.82·55-s + ⋯ |
| L(s) = 1 | − 1.30·3-s − 0.447·5-s − 0.324·7-s + 0.692·9-s − 0.852·11-s + 0.552·13-s + 0.581·15-s + 1.25·17-s − 1.30·19-s + 0.422·21-s + 0.790·23-s + 0.200·25-s + 0.399·27-s − 1.21·29-s + 1.32·31-s + 1.10·33-s + 0.145·35-s + 0.324·37-s − 0.718·39-s + 0.0609·41-s − 0.874·43-s − 0.309·45-s − 0.0456·47-s − 0.894·49-s − 1.63·51-s − 1.06·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6104877039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6104877039\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 2.25T + 3T^{2} \) |
| 7 | \( 1 + 0.859T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 1.99T + 13T^{2} \) |
| 17 | \( 1 - 5.17T + 17T^{2} \) |
| 19 | \( 1 + 5.67T + 19T^{2} \) |
| 23 | \( 1 - 3.79T + 23T^{2} \) |
| 29 | \( 1 + 6.51T + 29T^{2} \) |
| 31 | \( 1 - 7.37T + 31T^{2} \) |
| 37 | \( 1 - 1.97T + 37T^{2} \) |
| 41 | \( 1 - 0.390T + 41T^{2} \) |
| 43 | \( 1 + 5.73T + 43T^{2} \) |
| 47 | \( 1 + 0.312T + 47T^{2} \) |
| 53 | \( 1 + 7.73T + 53T^{2} \) |
| 59 | \( 1 + 8.04T + 59T^{2} \) |
| 61 | \( 1 + 0.491T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 - 6.08T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 4.45T + 79T^{2} \) |
| 83 | \( 1 + 7.68T + 83T^{2} \) |
| 89 | \( 1 - 4.34T + 89T^{2} \) |
| 97 | \( 1 - 5.68T + 97T^{2} \) |
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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
−7.88434672550393564594445724850, −6.97163261137638270763260583933, −6.32754790325927401727518662579, −5.82553022076558675724420062443, −5.06600930131088312559235726496, −4.53001494181760783274919535861, −3.52859219570045059208335602127, −2.80743871896028057833342270807, −1.49148968488895524925418598044, −0.42855031260435126063507534981,
0.42855031260435126063507534981, 1.49148968488895524925418598044, 2.80743871896028057833342270807, 3.52859219570045059208335602127, 4.53001494181760783274919535861, 5.06600930131088312559235726496, 5.82553022076558675724420062443, 6.32754790325927401727518662579, 6.97163261137638270763260583933, 7.88434672550393564594445724850
