| L(s) = 1 | + 2-s + 3-s + 4-s − 2.55·5-s + 6-s + 7-s + 8-s + 9-s − 2.55·10-s − 3.24·11-s + 12-s + 14-s − 2.55·15-s + 16-s + 3.40·17-s + 18-s + 2.85·19-s − 2.55·20-s + 21-s − 3.24·22-s − 8.54·23-s + 24-s + 1.52·25-s + 27-s + 28-s − 6.18·29-s − 2.55·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.14·5-s + 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.807·10-s − 0.979·11-s + 0.288·12-s + 0.267·14-s − 0.659·15-s + 0.250·16-s + 0.826·17-s + 0.235·18-s + 0.654·19-s − 0.571·20-s + 0.218·21-s − 0.692·22-s − 1.78·23-s + 0.204·24-s + 0.305·25-s + 0.192·27-s + 0.188·28-s − 1.14·29-s − 0.466·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 2.55T + 5T^{2} \) |
| 11 | \( 1 + 3.24T + 11T^{2} \) |
| 17 | \( 1 - 3.40T + 17T^{2} \) |
| 19 | \( 1 - 2.85T + 19T^{2} \) |
| 23 | \( 1 + 8.54T + 23T^{2} \) |
| 29 | \( 1 + 6.18T + 29T^{2} \) |
| 31 | \( 1 - 0.423T + 31T^{2} \) |
| 37 | \( 1 + 5.54T + 37T^{2} \) |
| 41 | \( 1 - 3.14T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 + 4.50T + 47T^{2} \) |
| 53 | \( 1 - 4.12T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 + 5.47T + 67T^{2} \) |
| 71 | \( 1 - 4.21T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 + 16.8T + 83T^{2} \) |
| 89 | \( 1 + 1.96T + 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−7.77448440409880088366217472259, −7.15465037313848863975219729532, −5.99839664684867970292191146167, −5.41559323758850949349930480095, −4.57392320744024706954975973969, −3.89508149228503824607364193601, −3.33981023526476858745438183164, −2.49260533850567831785645397877, −1.52045292117815995442945164803, 0,
1.52045292117815995442945164803, 2.49260533850567831785645397877, 3.33981023526476858745438183164, 3.89508149228503824607364193601, 4.57392320744024706954975973969, 5.41559323758850949349930480095, 5.99839664684867970292191146167, 7.15465037313848863975219729532, 7.77448440409880088366217472259
